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The axes of the joints of the structure are represented ly points in the diagram of the frame.
The link which connects two joints in the actual structure may be of any shape, but it the diagram of the frame it is represented by a straight line joining the point) representing the two joints.
If no force acts on the link except the two forces acting through the centres of the joints, these two forces must be equal and opposite, and their direction must coincide with the straight line joining the centres of the joints.
If the force acting on either extremity of the link is directed towards the other extremity, the stress on the link is called pressure and the link is called a strut. If it is directed away from the other extremity, the stress on the link is called tension and tho link is called a tie.
In this case, therefore, the only stress acting in a link is a pressure or a tension in the direction of the straight line which represents it in the diagram of the frame, and all that we have to do is to find the magnitude of this stress.
In the actual structure, gravity acts on every part of the link, but in the diagram we substitute for the actual weight of the different parts of the link, two weights which have the same resultant acting at the extremities of the link.
We may now treat the diagram of the frame as composed of links without weight, but loaded at each joint with a weight made up of portions of the weights of all the links which meet in that joint.
If any link has more than two joints we may substitute for it in the diagram an imaginary stiff frame, consisting of links, each of which has only two joints.
The diagram of the frame is now reduced to a system of points, certain pairs of which are joined by straight lines, and each point is in general acted on by a weight or other force acting between it and some point external to the system.
To complete the diagram we may represent these external forces as links, that is to say, straight lines joining the points of the frame to points external to the frame. Thus each weight may be represented by a link joining the point of application of the weight with the centre of the earth.
Bat we can always construct an imaginary frame having its joints in the lines of action of these external forces, and this frame, together with the real frame and the links representing external forces, which join points in the one frame to points in the other frame, make up together a complete self-strained system in equilibrium, consisting of points connected by Links acting by pressure or tension. We may in this way reduce any real structure to the case of a system of points with attractive or repulsive forces acting between certain pairs of these points, and keeping them in equilibrium.
The direction of each of these forces is sufficiently indicated by that of the line joining the points, so that we have only to determine its magnitude.
We might do this by calculation, and then write down on each link the pressure or the tension which acts in it
We should in this way obtain a mixed diagram in which the stresses are represented graphically as regards direction and position, but symbolically as regards magnitude.
But we know that a force may be represented in a purely graphical manner by a straight line in the direction of the force containing as many units of length as there are units of force in the force. The end of this line is marked with an arrow head to show in which direction the force acts.
According to this method each force is drawn in its proper position in the diagram of configuration of the frame. Such a diagram might be useful as a record of the result of calculation of the magnitude of the forces, but it would be of no use in enabling us to test the correctness of the calculation.
But we have a graphical method of testing the
equilibrium of any set of forces octing at a point. We draw in series a set of lines parallel and proportional to these forces. If these lines form a closed polygon the forces aro in equilibrium. We might in this "way form a series of polygons of forces, one for each joint of the frame. But in so doing we give up the principle of drawing the line representing a force from tho point of application of the force, for aU the sides of the polygon cannot pass through the same point, as the forces do.
We also represent every stress twice over, for it appears as a side of both the polygons corresponding to the two joints between which it acts.
But if we can arrange the polygons in such a way that the sides of °ny two polygons which represent tho same stress coincide with each other, we may form a diagram in which everystress is represented indirection and magnitude, though not in position, by a single line which is the common boundary of the two polygons which represent the joints at the extremities of the corresponding piece of the frame.
We have thus obtained a pure diagram of stress in which no attempt is made to represent the configuration of the material system, and in which every force is not only represented in direction and magnitude by a straight line, but the equilibrium of the forces at any joint is manifest by inspection, for we have only to examine whether the corresponding polygon is closed or not.
The relations between the diagram of the frame and the diagram of stress are as follows :—
To every link in tho frame corresponds a straight line in the diagram of stress which represents in magnitude and direction tho stress acting in that link.
To every joint of the frame corresponds a closed polygon in the diagram, and the forces acting at that joint are represented by the sides of the polygon taken in a certain cyclical order. The cyclical order of the sides of the two adjacent polygons is such that their common side is traced in opposite directions in going round the two polygons.
The direction in which any side of a polygon is traced is the direction of the force acting on that joint of the frame which corresponds to the polygon, and due to that Link of the frame which corresponds to the side.
This determines whether the stress of the link is a pressure or a tension.
If we know whether the stress of any one link is a pressure or a tension, this determines the cyclical order of the sides of the two polygons corresponding to the ends of the links, and therefore the cyclical order of all the polygons, and the nature of the stress in every link of the frame.
Definition of Reciprocal Diagram, When to every point of concourse of the lines in the diagram of stress corresponds a closed polygon in the skeleton of the frame, the two diagrams are said to be reciprocal.
The first extensions of the method of diagrams of forces to other cases than that of the funicular polygon were given by Rankine in his Applied Mechanic! (1857). The method was independently applied to a large number of cases by Mr W. P. Taylor, a practical draughtsman in the office of the well-known contractor Mr J. R Cochrane, and by Professor Clerk Maxwell in his lectures in King's College, London. In the Phil. Mag. tor 1864 the latter pointed out the reciprocal properties of the two diagrams, and in a paper on " Reciprocal Figures, Frames, and Diagrams of Forces," Tram. R. S. Edinburgh, vol xxvi (1870), he showed the relation of the method to Airy's function of stress and to other mathematical methods.
Professor Fleeming Jenkin hag given a number of apphV cations of the method to practice (Trans. R. S. JSdin., vol xxv)
Cremona(£<figure rtciprocha nella stattca grafica, Milan, 1872) has deduced the construction of reciprocal figures from tfao theory of the two components of a wrench as developed by Mbbius.
Culmann, in his Graphisclie Statii, makes great use of diagrams of forces, some of which, however, are not reciprocal.
M. Maurice Levy in his Slatique Graphique (Paris, 1874) lias treated the whole subject in an elementary but copious manner.
Mr R. H. Bow, C.E., F.R.S.E., in his work on The Economic* of Construction in relation to Framed Structures, 1873, has materially simplified the process of drawing a diagram of stress reciprocal to a given frame acted on by • system of equilibrating external forces.
Instead of lettering the joints of the frame, as. is usually done, or the links of the frame, as was the writer's custom, he places a letter in each of the polygonal areas inclosed by the links of the frame, and also in each of the divisions of surrounding space as separated by the lines of action of the external forces.
\7hen one link of the frame crosses another,-the point of apparent intersection of the links is treated as if it were a real joint, and the stresses of each of the intersecting links are represented twice in the diagram of stress, as the opposite sides of the parallelogram which corresponds to the point of intersection.
Fio. 2.—Diagram of Stress.
This method is followed in the lettering of the diagram of configuration (fig. 1), and the diagram of stress (fig. 2) of the linkwork which Professor Sylvester has called a quadruplane.
In fig. 1 the real joints are distinguished from the places where one link appears to cross another by the little circles O, P, Q, B, S. T, V.
The four links BSTV form a "contraparallelogram" in which ES - TV and RV - ST.
The triangles ROS, RPV, TQS are similar to each other. A fourth triangle (TNV), not drawn in the figure, would complete the quadruplanst The four points O, P, N, Q form a parallelogram whose angle POQ is constant and equal to x-SOR. The product of the. distances OP and OQ is aonstant,
The linkwork may be fixed at O. If any figure is traced by P, Q will trace the inverse figure, but turned round O through the constant angle POQ.
In the diagram forces Pp, Qq are balanced by the force Oo at the fixed point. The forces Pp and Qq are necessarily inversely as OP and OQ, and make equal angles with those lines.
Every closed area formed by the links or the external forces in the diagram of configuration is marked by a letter which corresponds to a point of concourse of lines in the diagram of stress.
The stress in the link which is the common boundary of two areas is represented in the diagram of stress by the line joining the points corresponding to those areas.
When a link is divided into two or more parts by lines crossing it, the stress in each part is represented by a different line for each part, but as the stress is the same througbont the link these lines are all eqnal and parallel. Thus in the figure the stress in RV is represented by the four equal and parallel lines HI, FG, DE, and AB.
If two areas have no part of their boundary in common the letters corresponding to them in the diagram of stress are not joined by a straight line. If, however, a straight line were drawn between them, it would represent in direction and magnitude the resultant of all the stresses in the links which are cut by any line, straight or curved, joining the two areas.
For instance the areas F and C in fig. 1 have no common boundary, and the points F and C in fig. 2 are not joined by a straight line. But every path from the area F to the area C in fig. 1 passes through a series of other areas, and each passage from one area into a contiguous area corresponds to a line drawn in the diagram of stress. Hence the whole peth from F to C in fig. 1 corresponds to a path formed of lines in fig. 2 and extending from F to C, and the resultant of all the stresses in the links cut by the path is represented by FC in fig. 2.
Automatic Description of Diagrams.
There are many other kinds of diagrams in which the two co-ordinates of a point in a plane are employed to indicate the simultaneous values of two related quantities.
If a sheet of paper is made to move, say horizontally, with a constant known velocity, while a tracing point is made to move in a vertical straight line, the height varying as the value of any given physical quantity, the point will trace out a curve on the paper from which the value of that quantity at any given time may be determined.
This principle is applied to the automatic registration of phenomena of all kinds, from those of meteorology and terrestrial magnetism to the velocity of cannon-shot, the vibrations of sounding bodies, the motions of animals, voluntary and involuntary, and the currents in electric telegraphs.
In Watt's indicator for steam engines the paper does not move with a constant velocity, but its displacement is proportional to that of the piston of the engine, while that of the tracing point is proportional to the pressure of the steam. Hence the. co-ordinates of a point of the curve traced on the diagram represent the volume and the pressure of the steam in the cylinder. Tho indicatordiagram not only supplies a record of the pressure of the steam at each stage of the stroke of the engine, but indicates the work done by the steam in each stroke by the area inclosed by the curve traced on the diagram.
The indicator-diagram was invented by James Watt as a method of estimating the work done by an engine. It was afterwards used by Clapeyron to illustrate the theory of boat, and this use of it wag greatly developed by Rankine in his work on the steam engine.
The use of diagrams in thermodynamics has been very completely illustrated by Fiof. J. Willard Gibbs (Conneo
ticut Acad. Sci., vol. iii.), but though his methods throw much light on the general theory of diagrams as a method of study, they belong rather to thermodynamics than to the present subject. (j. a x.)
DIALLING, sometimes called gnomonics, is a branch of applied mathematics which treats of the construction of sun-dials, that is, of those instruments, either fixed or portable, which determine the divisions of the day by the motion of the shadow of some object on which the sun's ray3 fall
It must have been one of the earliest applications of a knowledge of the apparent motion of the sun ; though for a long time men would probably be satisfied with the division into morning and afternoon as marked by sun-rise, sun-set, and the greatest elevation.
History.—The earliest mention .of a sun-dial is found in Isaiah xxxviii. 8: "Behold, I will bring again the shadow of the degrees which is gone down in the sun-dial of Ahaz ten degrees backward." The date of this would be about 700 years before the Christian era, but we know nothing of the character or construction of the instrument
The earliest of all sun-dials of which we have any certain knowledge was the hemicyele, or hemisphere, of the Chaldean astronomer Berosus, who probably lived about 310 B.c. It consisted of a hollow hemisphere placed with its rim perfectly horizontal, and having a bead, or globule, fixed in any way at the centre. So long as the sun remained above the horizon the shadow of the bead would fall on the inside of the hemisphere, and the path of the shadow during the day would be approximately a circular arc. This arc, divided Into twelve equal parts, determined tweKo equal intervals of time for that day. Now, supposing this were done at the time of the solstices and equinoxes, and on as many intermediate days as might be considered sufficient, and then curve lines drawn through the corresponding points of division of the different arcs, the shadow of the bead falling on one of these curve lines would mark a division of time for that day, and thus we should have a sun-dial which would divide each period of daylight into twelve equal parts.
These equal parts were called temporary hours; and, since the duration of daylight varies from day to day, the temporary hours of one day would differ from those of another; but this inequality would probably be disregarded at that time, and especially in countries where the variation between the longest summer day and the shortest winter day is much less than in our climates. •
The dial of Berosusremained in use for centuries. The Arabians, as appears from the work of Albategnius, still followed the same construction about the year. 900 A.d. Four of these dials have in modem times been found in Italy. One, discovered at Tivoli in 1716, is supposed to have belonged to Cicero, who, in one' of his letters, says that he had Bent a dial of this kind to his villa near Tusculum. The second and third were found in 1751— one at Castel-Nuovo, and the other at Rignano; and a fourth was found in 1762 at Pompeii G. H. Martini, the author of a dissertation in German on the dials of the ancients, says that this dial was made for the latitude of Memphis; it may therefore be the work of Egyptians, perhaps constructed-in the school of Alexandria.
It is curious that no sun-dial has been found among the antiquities of Egypt, and their sculptures give no indication of any having existed. It has, however, been supposed that the numerous obelisks found everywhere were erected in honour of the sun and employed as gnomons.
Herodotus has recorded that tho Greeks derived from the Babylonians the use of the gnomon, but the •_Tc.it pro* mado by the Creeks in geometry enabled them in times to construct dials of great complexity, some of which remain to us, and are proofs, not only of extensive knowledge, but also of great ingenuity.
Ftolemy's Synlaxis treats of the construction of dials by means of his analemma, an instrument which solved a variety of astronomical problems. The constructions given by him were sufficient for regular dials, that is, horizont-vl dials, or vertical dials facing east, west, north, or south, and these arc the only ones he treats of. It is certain, however, that the ancients were able to construct declining dials, as is shown by that most interesting monument of ancient gnomonics—the Tower of the Winds—which is still in existence at Athens. This is a regular octagon, on the faces of which the eight principal winds are represented, and over them eight different dials—four facing the cardinal points and the other four facing the intermediate directions. The date of the dials is long subsequent to that of the tower; for Vitruvius, who describes the tower in the sixth chapter of his first book, says nothing about the dials, and as he has described all the dials known in his time, we must believe that the dials of the tower did not then exist. The tower and its dials are described by Stuart in his Antiquities of Athens. The hours are still the temporary hours, or, as the Greeks called them, hectemoria.
As already stated, the learning and ingenuity of the Greeks enabled them to construct dials of various forms— among others, dials of suspension intended for travellers; but these are only spoken of and not explained; they may have been like our ring-dials.
The Romans were neither geometers nor astronomers, and the science of gnomonics did not flourish among them. The first sun-dial erected at Rome was in the year 290 B.c, and this Papirius Cursor had taken from the Samnites. A dial which Valerius Messala had brought from Catania, the latitude of which is five degrees less than that of Rome, was placed in the forum in the year 261 &a The first dial actually constructed at Rome was in the year 164 B.c, by order of Q. Marcius Philippus, but, as no other Roman has written on gnomonics, this was perhaps the work of a foreign artist If, too, we remember that the dial found at Pompeii was made for the latitude of Memphis, and consequently less adapted to its position than that of Catania to Rome, we may infer that mathematical knowledge was not cultivated in Italy.
The Arabians were much more successful. They attached great importance to gnomonics, the principles of which they had learned from the Greeks, but they greatly simplified and diversified the Greek constructions. One of their writers, Abul-Hassan, who lived about the beginning of the 13th century, taught them how to trace dials on cylindrical, conical, and other surfaces. He even introduced equal Or equinoctial hours, but the idea was not supported, and the temporary hours alone continued in use.
Where or when the great and important step already conceived by Abul-Hassan, and perhaps by others, of reckoning by equal hours was generally adopted cannot now be determined. The .history of gnomonics from the 13th to the beginning of the 16th century is almost a blank, and during that time the change took place. We
VIL — so
can tee, however, that the change would necessarily follow the introduction of clocks and other mechanical methods of measuring time; for, however imperfect theso were, the hours they marked would be of the same length in summer and in winter, and the discrepancy between these equal hours and the temporary hours of the sun-dial wonld soon be too important to be overlooked. Now, we know that a balance clock was put up In the palace of Charles V. of France about the year 1370, and wo may reasonably supposo that tho new sun-dials came into general use during the 14th and 15th centuries.
Among the earliest of the modem writers on gnomonics must be named Sebastian Munster, a cordelier who published his Horologiographia&t BaseHn 1531. He gives a number of correct rules, but without demonstrations. Among his inventions was a moon-dial,1 but this docs not admit of much accuracy.
During tho 17 th century dialling was discussed at great length by all writers on astronomy. Clavius devotes a quarto volume of 800 pages entirely .to the subject This was published in 1612, and may be considered to contain all that was known at that time.
In the 18th century clocks and watches began to supersede sun-dials, and these have gradually fallen into disuse except as an additional ornament to a garden, or in remote country districts where tho old dial on the church tower still serves as an occasional check on the modern clock by its side. The art of constructing dials may now be looked upon as little more than a mathematical recreation.
General Principles.—The diurnal and the annual motions of the earth are the elementary astronomical facts on which dialling is founded. That the earth turns upon its axis uniformly from west to east in 24 hours, and that it is carried round tho sun in one year at a nearly uniform rate, is, we know, the correct way of expressing these facts. But the effect will be precisely the same, and it will suit our purpose better, and make our explanations easier, if we adopt the ideas of the ancients, of which our senses furnish apparent confirmation, and assume the earth to be fixed. Then, the sun and stars revolve round the earth's axis uniformly from east to west once a day,—the sun lagging a little behind the stars, making its day some 4 minutes longer, so that at the end of the year it finds itself again in the same place, having made a complete revolution of the heavens relatively to the stars from west to east
The fixed axis about which all these bodies revolve daily is a line through the earth's centre; but the radius of the earth is so small, compared with the enormous distance of the sun, that, if we draw a parallel axis through any point of the earth's surface, we may .safely look on that as being the axis of the celestial motions. The error in the case of the sun would not, at its maximum, that is, at 6 A.m. and 6 P.m., exceed half a second of time, and at noon would vanish.
An axis so drawn is in the plane of the meridian, and points, as we know, to the pole,—its elevation being equal to the latitude of the place.
The diurnal motion of the stars is strictly uniform, and so would that of the sun be if the daily retardation of about 4 minutes, spoken of above, were always the same. But this is constantly altering, so that the time, as measured by the sun's motion, and also consequently as measured by a sun-dial, does not move on at a strictly uniform pace. This irregularity, which is slight, would be of little consequence in the ordinary affairs of life, bnt clocks and
1 In one of the Court* of Queen's College, Cambridge, then la an elaborate sun-dial dating from the end of the 17th or beginning of the 18th century, and around it a aeries of nnmben which make It available as a moon-dial when the moon's age la knows.
watches being mechanical measures of time could not, except by extreme complication, be made to follow this irregularity, even if desirable, which is not the case.
The clock is constructed to mark uniform time in such wise that the length of tho clock day shall be the average of all the solar days in the year. Four times a year the clock and the sun-dial agree exactly ; bnt the sun-dial, now going a little slower, now a little faster, will be sometimes behind, sometimes before the clock—the greatest accumulated difference being about 16 minutes for a few days in November, but on the average much less. The four days on which the two ugreo are April 15, June 15, September 1, and December 24.
Clock-time is called mean time, that marked by the sundial is called apparent time, and the difference between them is the equation of time. It is given in most calendars and almanacs, frequently under the heading *' clock slow," "clock fast" When the time by the sun-dial is known, the equation of time will at once enable us to obtain the corresponding clock time, or vice versa.
Atmospheric refraction introduces another error, by altering the apparent position of tho sun ; but the effect is too small to need consideration in the construction of an instrument which, with the best workmanship, does not after all admit of very great accuracy.
The general principles of dialling will now be • readily understood. The problem before us is the following:— A rod, or style, as it is called, being firmly fixed in a direction parallel to the earth's axis, we have to find how and where points or lines of reference must ba traced on some fixed surface behind the style, so that when the shadow of the style falls on a certain one of these lines we may know that at that moment it is solar noon,—that is, that the plane through the style and through the sun then coincides with tho meridian; again, that when the shadow reaches the next line of reference, it is 1 o'clock by solar time, or, which comes to the same thing, that the above plane through tha style and through the sun has just turned through the twenty-fourth part of a complete revolution; and so on for the subsequent hours,—the hours before noon being indicated in a similar manner. The style and the surface on which these lines are traced together constitute the dial.
The position of an intended sun-dial having been selected —whether on church tower, south front of farm-stead, or garden wall—the surface must be prepared, if necessary, to receive the hour-lines.
The chief, and in fact the only practical difficulty will be the accurate fixing of the style, for on its accuracy the value of the instrument depends.
It must be in the meridian plane, and must make an angle with the horizon equal to the latitude of the place. The latter condition will offer no difficulty, but the exact determination of the meridian plane which passes through the point where the style is fixed to the surface is not so simple. We shall, further on, show how this may be done; and, in the meantime, we shall'assume that we have found the true position, and have firmly fixed the style to the dial and secured it there by cross wires, or by. other means. The style itself will be usually a strong metal- wire whose thickness may vary with circumstances; and when we speak of the shadow cast by the style it must always be understood that the middle line of the thin band of shade is meant
The point where the style meets the dial is called the centre of the dial. It is the centre from which all the hour-lines radiate.
The position of the xn o'clock line is the most important to determine accurately, since all the others are usually .made to depend on this one. We cannot trace it correctly on the dial until the style has been itself accurately fixed in its proper place, as will be explained hereafter. When that is done the xn o'clock line will" be fonnd by the intersection of the dial surface with the vertical plane which contains the style; and the most simple way of drawing it on the dial will be by suspending a plummet from some point of tho style whence it may hang freely, and waiting until the shadows of both style and plumb line coincide on the dial. This single shadow will be the xn o'clock
In one "class of dials, namely, all the vertical ones, the xn o'clock line is simply the vertical line from the centre; it can, therefore, at once be traced on the dial face by using a fine plumb line.
The xn o'clock line being traced, the easiest and most accurate method of tracing the other hour lines would at the present day when good watches are common, be by marking where the shadow of the style falls when 1, 2, 3, 4c., hours have elapsed since noon, and the next morning by the same means the forenoon hour lines could be traced; and in the same manner the hours might be subdivided into halves and quarters, or even into minutes.
But formerly, when watches were not, the tracing of the t, n, m, dec. o'clock lines was done by calculating the angle which each of these lines would make with the xn o'clock line. Now, except in the simple cases of a horizontal dial or of a vertical dial facing a cardinal point, this* would require long and intricate calculations, or elaborate geometrical constructions, implying considerable mathematical knowledge, but also introducing increased chances of error. The chief source of error would lie in the uncertainty of the data; for the position of the dial-plane would have to be found before the calculations began,— that is, it would be necessary to know exactly by how many degrees it declined from the south towards the east or west, and by how many degrees it inclined from the vertical The ancients, with the means at their disposal, could obtain these results only very roughly.
Dials received different names according to their position :—
Horizontal dials, when traced on a horizontal plane;
Vertical dials, when on a vertical plane facing one of the cardinal points;
Vertical declining dials, on a vertical plane not facing a cardinal point;
Inclining dials, when traced on planes neithei vertical nor horizontal (these were further -distinguished as reclining when leaning backwards from an observer, proclining when leaning forwards);
£qu> noctial dials, when the plane is at right angles to the earth's axis, &c dec
Vie shall limit ourselves to an investigation of the simplest and most usual of these cases, referring the reader, for further details, to the later works given at the end of thin article.
thai Construction.—A very correct view of the problem of dial construction may be obtained as follows :—
Conceive * transparent cylinder (fig. 1) having an axis AB parallel to tho axis of the earth. On the surface of the cylinder let equidistant generating lines ho traced 15* apart, one of them XII.. XII being in the meridian plane through AB, and the others I...I, II...ii, Ac, following in the order of tho sun's motion.
Then the shadow of the line AB will obviously fall on the line XII.. JCU at apparent poon, on the line I...I at one hour after noon, on n...n at two hours after noon, and so on. If now the cylinder be cut by any plane MN representing tho plana on which the dial is to be traced, the shadow of AB will be intercepted by this plane, and fall on the lines Axil, Al, An, Ac , The construction of the dial consists in determining the angles made by Al An, Ac. with Axil; the line Ax« itself, being In the vertical plane through AB, may be supposed known.
For the purposes of actual calculation, perhaps a trans
the former horizontal, the other perpendicular to the axis Pp, and therefore coinciding with tho plane of the equator. Let EZ be vertical, then the circle QZP will be the meridian, and, by its intersection A with the horizontal will determine the xn o clock line EA. Kext divide the equatorial circle QMa into 24 equal parts at, be, cd, Ac ... of 16 each, beginning from the meridian Pa, and through the various points of division and the poles draw the great circles Tbp, Pep, kc . . . These will exactly correspond to the equidistant generating lines on the cylinder in the previous construction, and the shadow of the stylo will fall on these circles after successive intervals of 1, 2, 8, fee. hours from noon.. If they meet the horizontal in the points B, 0, D, kc, then SB, EC, ED,kc ... . will be the I, H, m, Ac, hour lines required ; and the problem of the horizontal dial consists in calculating the angles which these lines make with the xn o'clock line EA, whose position is known. The spherical triangles PAB, PAO, kc, enable us to do this readily. They are all right-angled at A, tiie side PA is the latitude of the place, and the angles APB, APC, Ac, are respectively 15", SO*, Ac, then
tan, AB—tan. 16* sin. latitude, tan. AC = tan. 80* sin. latitude, Ac, Ac.
These determine the sides AB, AO, kc that is, the angles AEB, AEO, Ac, required.
For examples, let us find the angles made by the I o'clock line at the following places—Madras, London, Edinburgh, and Hammer test (Norway).