Thermometer BA,, which was heated about fifty times during Novem ber 1902 in electric furnaces up to 1050°, and again during April and May 1903 to similar temperatures for prolonged periods, appears to be hardly perceptibly affected by it, no certain change of FI occurring during the period February 12 to August 18 covered by the later experiments, and certainly no variation of the zero of 1° C.

To see if the small lack of homogeneity of the wire as shown by the properties of the different thermometers was due to the treatment it had received during the successive adjustments of FI, a new thermometer, named BA,, was made up of wire taken from the inner end of the same reel as the other six. No attempt was made at adjustment of its FI, which was found after thorough annealing to be 100-022 box units.

The 8 was found to be 1.506, an intermediate value. The wire was then unwound from the mica frame and suspended freely in air between the ends of the leads, and a current of 2 ampères, which was sufficient to maintain it at about 1400° C., was passed for about 2 hours.

Owing to the volatilisation of a considerable quantity of platinum from the wire, a large increase in the FI was found, as was expected, but the & remained unchanged, though a rise in R was recorded amounting to 1 part in 1000.

Ro

In order to make certain that the differences observed were not due to defective insulation in the thermometers, the insulation resistance between the thermometer and compensator leads of each of the thermometers was measured by a direct deflection method, and found to be in no case less than 700,000 ohms at any temperature between 0° and 1000° for BA[ and BA, and 0° and 500° for BA, and BA,. Some experiments were also made on an imitation platinum thermometer having its coil wound on mica of standard quality, but cut at the lower end into two parts.

Although the insulation from one part to another was practically infinite at all temperatures, when only platinum and mica were present in the heated part of the porcelain tube, the introduction of a small piece of clean copper wire into the hot space near the bulb was sufficient after some time to lower the insulation, even at only about 800° C., to a few thousand ohms. The cause of the differences between the individual thermometers does not, therefore, appear to be leakage.

Neither does the cause of the small differences in values of & found lie in the method of taking the sulphur point, as the same apparatus was used in the same way for all the experiments. The sulphur is now boiled in an arrangement similar to Callendar and Griffiths's well-known pattern, except that, to avoid the necessity of removing the tube at each reheat after the sulphur has crystallised, the glass boiling-tube is replaced by one of thin weldless steel, brazed with spelter into a rather wider endpiece of thick iron tubing, which is exposed to the direct flame of the large bunsen used for heating. The level of the liquid sulphur is always maintained at least 2 inches above the bottom plate of the apparatus, and the upper level of the vapour to a definite position, which can be seen through mica windows in the upper part of the neck. Under these conditions no measurable superheating of the vapour has ever been observed, and a comparison of the sulphur points obtained with this form of apparatus with those got in the older one, with glass boiling-tube, reveals no measurable systematic difference.

For the boiling point of sulphur under normal pressure in latitude 45′

dp

Callendar and Griffiths's old value, 444°53 C., has been retained, as dt was also the figure deduced by them from Regnault's experiments for for sulphur, namely, '082° C. per mm., although it has been shown independently, by Chree and by Harker and Chappuis, that this value for the variation is considerably too small. It is hoped that a redetermination of this constant for pressures between 700 and 800 mm. will shortly be undertaken in the thermometric laboratory.

APPENDIX IV

The following table gives the resistance at a temperature of 60° Fahr. (15°55 C.) of a wire of pure annealed copper 1 metre in length, having a mass of 1 gramme, as deduced from the most recent determinations.

In making the reductions, the values for the temperature coefficient and for the density given by the author, have been used.

Table giving Resistance at 60 Fahr, of a Wire of Pure Annealed Copper, such that 1 metre weighs 1 gramme.

On the Use of Vectorial Methods in Physics.

By Professor O. HENRICI, Ph.D., F.R.S.

[Ordered by the General Committee to be printed in extenso.]

HAVING been engaged for over thirty years in teaching mathematics, chiefly to engineering students, I have always had much sympathy with them. They have to consider mathematics as a tool to help them in their work; abstract reasoning is in many cases a horror to them. At school they have most likely been treated as duffers, unable to learn mathematics; but if the subject is led up to through concrete examples, everything becomes alive and full of interest to them. It is for such men as these that I speak primarily, not for mathematicians. It is for them that I advocate the more general use of vectors and their introduction into the school curriculum; because vectors give the most natural mathematical expressions for many quantities in dynamics and physics, and their introduction helps in the study of these subjects and in obtaining clear views. of the quantities dealt with.

Who

The very invention of vectors is due to the needs of dynamics, and he who first represented a force by a directed line is their inventor. this was seems to be unknown; Newton was the first who clearly stated the 'Parallelogram of Forces.' Since his time vectors have always been used in dynamics, although the name 'vector' was only introduced by Hamilton.

That this representation of a force by a vector is natural no one will dispute, but only the addition of vectors (composition and decomposition of forces) was in use until Hamilton and Grassmann almost simultaneously, In reducing Professor Fleming's result, the density has been taken as 8.91 grammes per c.c.

and from very different points of view, developed a calculus of vectors by defining their products. The applications of this new calculus to physics (including dynamics) remained long restricted to a few of their followers. It was, however, before their time that Faraday by his 'Lines of Force' and Fields of Forces' gave a purely geometrical representation of the phenomena of electricity and magnetism. Their analytical expression requires vectors. The first who recognised this was Clark Maxwell, and there can be little doubt that his success in putting Faraday's ideas into analytical form was greatly due to his knowledge of quaternions. His statements in the preface to, and in the preliminary chapter of, his 'Electricity and Magnetism' are in this respect of great interest. I quote from the latter: 'But for many purposes in physical reasoning, as distinguished from calculation, it is desirable to avoid explicitly introducing the Cartesian co-ordinates, and to fix the mind at once on a point of space instead of its three co-ordinates, and on the magnitude and direction of a force instead of its three components. This mode of contemplating geometrical and physical quantities is more primitive and more natural than the other, although the ideas connected with it did not receive their full development till Hamilton made the next great step in dealing with space by the invention of his calculus of quaternions.

'As the methods of Descartes are still the most familiar to students of science, and as they are really the most useful for purposes of calculation, we shall express all our results in the Cartesian form. I am convinced, however, that the introduction of the ideas, as distinguished from the operations and methods of quaternions, will be of great use to us in the study of all parts of our subject, and especially in electro-dynamics, where we have to deal with a number of physical quantities, the relations of which to each other can be expressed far more simply by a few words of Hamilton's than by the ordinary equations.'

He goes on: 'One of the most important features of Hamilton's method is the division of quantities of scalars and vectors.'

I have heard these words quoted as a proof that Maxwell was altogether in favour of Cartesian methods, and against quaternions and vectors. But this is wrong so far as vectors are concerned. In fact, the ideas which he took from Hamilton are chiefly two-first, vectors; and second, the classification of physical quantities into scalars and vectors. It is well known that he attached very great importance to the latter in connection with the theory of 'Dimensions.'

This classification has been carried further by Clifford. Certain vector-quantities require position for their full specification; Clifford says such a quantity is localised,' and calls a localised vector a 'rotor.' 2 Forces, spins, momentum, are examples. There are also localised scalars like mass and energy.

In connection with this subject the enforced absence, due to ill-health, of Mr. Williams is much to be regretted. He has continued his valuable work of the Theory of Dimensions, and has lately taken 'position' into account. It was hoped that he would communicate some of his recently obtained results at this meeting, and thus bear witness to the importance of vectors in this direction.

See his paper, 'Classification of Physical Quantities,' Proc. Lond. Math. Soc., vol. iii. p. 224.

2 Professor Joly has pointed out to me that Hamilton has also considered these. In his unpublished papers he calls them tractors.'