It has been said that, in political matters, a lie, uncontradicted for twentyfour hours, is as good as a truth. How long a false theory in physics requires to obtain this prescription has not been stated, but thinking it high time the prevalent opinion on the pendulum experiments should be further inquired into, I send you with much diffidence a few remarks. The English mathematicians profess to have derived their solution of the problem from the French. It does not appear that any men of note here have published original or complete investigations themselves. It is to be regretted that men so able to form original theories themselves as Professors Young and Powell, should have been contented with merely illustrating those of others. If it should appear, on their further investigations, that the theory is unsound, they will participate in this regret. Meantime, without any pretensions to be able to form another complete theory, I am anxious to state the views I take of theirs, which, if they have no effect in modifying it, may make some approach to a more popular appreciation of the phenomena.

At the outset I wish to guard myself against being supposed in the least degree to call in question the reality of the experiments. I think they most assuredly form the basis of a demonstration of the rotation of the earth of a highly interesting character, and peculiarly fitted to strike the popular mind. The doubts expressed upon the subject from various quarters, are obviously the result of most defective arrangements for performing the experiments. No such doubts have arisen from the experiments performed in this place under the direction of your occasional correspondent, Mr. Uriah Clarke-none of the apparent anomalies

having arisen. In fact, the experiment is a new one, requiring great care to adjust the parts; but when the necessary care is taken, the result is uniformly decisive of the effect being caused by the rotation of the earth.

The theory we wish to examine is thus propounded:

Mr. Silvester, in his letter to the Times, April 25th, inserted in your No. 1447, says "M. Binet's investigation leaves nothing to be desired in point of vigour of demonstration; the same result has been obtained, after a more compendious method, by two English analysts, the correctness of which I can attest. For the sake of those thinkers who form an intermediate class between those who are incapable of any proof except what appeals directly to the senses, and the elevated few who can comprehend the form of an analytical investigation, I offer a brief recapitulation of the argument only hinted at in my former letter.

"At the pole it is obvious the plane of vibration remains fixed in space, and therefore appears to revolve at the rate of 15° per hour. At the equator the 'law of sufficient reason' shows there can be no apparent rotation either way.

As regards places intermediate

A paper explanatory of the experiments referred to, by our esteemed friend Mr. Clarke, appeared in the Leicester Mercury about six weeks ago. We meant to have republished it in our pages, but it has been excluded by a pressure of other matters till its appearance now would be out of season. According to the theoretical rule for ascertaining the apparent deflection-namely, 15° per hour at the pole, and for other places as the sine of the latitude-the deflection should be somewhat less than 12° for the latitude of Leicester; but Mr. Clarke found, "from actual experiments many times repeated," that it is more than 12°. This discrepancy is supposed by Mr. Clarke to arise from the theoretical calculations having been made on the supposition of the earth being a spherical instead of a spheroidal body.

between the pole and the equator, a rough but substantially correct view of the phenomenon may be reasoned out thus:By a process known to geometers, the rotation of the earth about its pole may be supposed to be replaced by two simultaneous rotations about two ideal poles, one running straight up under the place of observation, the other passing through the earth's centre at right angles to the former. This latter pole will produce no effect. The principal, and I may say total observed effect, will be due solely to the rotation about the ideal pole, which is on a line with the point of suspension, and the rate of its vibration is the entire and true rate diminished as the sine of the latitude to unity. The motion is the result of a rational and mathematical investigation in which two things have to be considered-the motion of the earth and the motion of the plane itself, if the mode of explanation be attempted to be kept up; and the difference of these two motions will become apparent to observation."

Now as at the pole the apparent motion is due to the invariability of the pendulum's plane, coupled with the earth's motion, at every other place the same effect would be produced in a complete revolution in twenty-four hours if the plane remained invariable, and therefore any variation in the time of a revolution could only be effected by an actual motion of the plane,-a mode of explanation, however, which Mr. Silvester seems to deprecate.

To say nothing about the easy way in which Mr. Silvester disposes of one of the two ideal rotations into which, for solving the problem, he supposes the rotation of the earth to be resolved, nothing can be clearer than that it is impossible to neglect either, unless it has its whole effect expended, in verily and indeed causing the plane of vibration to move in the same direction. Whether any such effect is due to the rejected rotation, and if so, in what manner it is accomplished, all the writers upon this subject are entirely silent; and yet they all follow M. Binet in making this artifice of geometry first used by Euler the foundation of their solution. It is of course difficult for a person of small attainments in calculations to say the process is inapplicable to the question altogether, but it is not too much to ask

those better informed to point out some reasons why they suppose it to be applicable. The very loose terms in which Mr. Silvester states his proposition are such as not to create any great confidence in it; and your very intelligent correspondent, "S. Y.," who has on former occasions wielded no weak weapons against the undue pretensions of the mere mathematicians, is so struck by the insufficiency of the ground for first resolving the real rotation into two ideal ones, thus treating both as real, and afterwards arbitrarily rejecting one of them as having no effect, that he seems to doubt the reality of the phenomenon altogether. However, in the imperfect report of Professor Powell's lecture, we are told that the Astronomer Royal agrees with him in opinion that the exact determination of the direction of the plane cannot be made on any general considerations, but must be the result of detailed mathematical investigation.

Let us therefore return to the inquiry. At the pole the plane of vibration continues invariable-it maintains its first position, 1st, in direction; 2nd, in its relation to the plane of rotation; 3rd, in the horizontal direction of the line joining the centres of the ball at the two extreme distances. At the equator the plane of vibration is equally invariable in direction; for if the pendulum be set vibrating north and south, although the plane will revolve round with the earth, the direction of the plane north and south will be unchangeable. In like manner, if set vibrating east and west, the plane will remain unchanged, though the direction of the line joining the centres of the ball at the two extremes will vary through the whole circle. At any intermediate latitude, the plane will vary in inclination from the perpendicular on the plane of rotation to an angle equal to the latitude of the plane; and the line joining the centres of the ball at the two extremes will vary from a parallel with the plane of rotation to an inclination equal to the same angle; but it is not equally obvious, though it may be true, that the direction of the plane varies at all. Now this is an important consideration; for from the fact of the table under the pendulum maintaining always the same position with respect to the axis of the pendulum, and consequently with relation to the line where the plane

of the pendulum cuts it, this direction of the plane is the only quantity affecting the problem at all; and if that is invariable, the table under it will make a complete revolution in twenty-four hours in every part of the earth except at the equator, however much the other conditions may vary. Before proceeding to investigate in what way this direction of the plane may be liable to vary, let us see how this view of the matter affects the theory under examination. We have before concluded that if Euler's process is applicable to this question, it will be by one of the rotations having its whole effect expended in making plane of vibration really move in the same direction as the earth. We can now substitute the direction of the plane instead of the plane itself.

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Now as in travelling from the pole to the equator, this theory assumes that the quantity of apparent motion of the pendulum, as caused by the effective rotation, is constantly decreasing, the effect of the other rotation is continually increasing till, in approaching the equator, it will at last become indefinitively near to a complete revolution. How comes it, then, when it arrives at the equator it will, as we have just seen, have no effect whatever, the direction of the plane there remaining absolutely unchangeable?

With great respect for Professor Young, I consider the explanation given by him neither true nor consistent with the theory of M. Binet. The professor not only considers the plane of vibration at all latitudes invariable, but he even considers the line joining the centres of the ball at the two extremes as absolutely

it must still be horizontal; and as the horizontal plane has changed the direction of its inclination, the line has no longer the inclination of the tangent to the meridian it had at first, and an addition will have to be made to the angle of its motion of a quantity due to the change of its inclination. In its motion to each succeeding meridian, the angle to be added will be an increasing quantity; so that, on his own supposition thus corrected, when the earth has revolved 90°, the motion will have accomplished 90° also.

There is a mode of viewing Professor Young's theory which would render it intelligible, but which shows its utter inapplicability to solve the problem. By supposing the line to retain the first inclination, and not to follow the horizontal plane, it would truly have the motion he ascribes to it, and would form a double cone during one revolution of the earth, each exactly similar to the cone formed by the prolonged tangents to the meridians; and in this manner might be considered as making a complete revolution without going through 360°: but that the professor could conceive this as supporting Mr. Silvester's views, which he considered abundantly satisfactory, I am at a loss to conceive.

Let us now, however, endeavour to investigate, in an ordinary manner suited to common minds, by what means the direction of the plane at any latitude could be made to vary, and in the outset let us divest ourselves of the prevalent, though vague, notion that some of the earth's motions could of themselves effect this object. No truth in physics is better established than that any num

tion along any one assumed meridian. This appears a strange misconceptionbut his whole theory is built upon it. He says this line, during the motion of the earth from one meridian to another, will make an angle, not of the distance of the meridians, but of their inclinations at the apex of a cone formed by a prolongation of the tangents at the latitude in question. Now as a mathematical deduction from the immediate premises, this is unquestionable; but these premises are not the conditions of the problem. When the horizontal line, as he calls it, has arrived at the second position, and assumed a different direction,

perfectly independent of each other, it is the very principle of the second law of motion which it is too late in the progress of science to expect to see repealed. The only way in which one motion of a body can be instrumental in causing any alteration of another motion, is by bringing it within the action of a force, or by altering its relation to a force to which it is incident. In this instance, the motion of the earth can bring the pendulum within the sphere of no new force. Can it, therefore, alter its relation to the force of gravity to which it is already incident? Let us In the oscillations of a pendulum the ball is alternately on each side of the axis, in an exactly symmetrical manner; any effect, therefore, that can be produced on the ball in its passage from the axis to the extreme distance and back again, on one side, would be exactly balanced by a corresponding effect on the other side, and, as far as the present question is concerned, the whole effect on each side might be considered as concentrated in that on the ball when at each extreme. It will not, perhaps, therefore, involve any error if we suppose the force of gravity on the pendulum represented by a line from the centre of the earth to the ball of the pendulum at each of the two extremes. And if we consider the plane of the pendulum, and its axis extended to the centre of the earth within these two lines, we shall be able easily to see what effect any alteration of its position may be likely to cause. In the first place, this extended plane will at all times be in the section of a great circle of the sphere, that at all times when at rest gravity can have no effect whatever in causing any deflection of the plane, its action on the plane being in the direction of it, and symmetrical on each side its axis. Moreover, it might to some minds be sufficient to say that as the rotation of the earth in altering the position of this plane acts on a single point, that of suspension of the pendulum, or the extremity of the axis prolonged from the centre, the force of gravity from the centre can have no effect whatever in causing the plane to deflect or twist in any direction. But a little detailed consideration may make it clearer; first, the motion of the point of suspension is in a circle on the outside of a sphere, which motion may obviously be resolved into two motions, -one in a great circle in the direction of the plane itself, and the other in a great circle at right angles to the former. And this would in fact be true in its motion from one point to another, independent of the two points lying in a circle. Now, the motion of the plane in its own direction obviously

never alters its relation to the central force, and its motion at right angles to it, as it is in a great circle, never alters its position as regards the centre, that remaining perfectly symmetrical to it every moment. Therefore the united action, simultaneously, in these two

directions cannot have any effect. Consequently, if the assumptions I have started with are allowable, the direction of the plane of vibration at any latitude is absolute, invariable. If the assumptions are not allowable to the extent I have taken them, the simplicity of the process, perhaps, may make the corrections that would be necessary easy to be understood, and I may be informed by others more competent to such inquiries, what the amount due to such corrections is.

Some general considerations seem to strengthen these conclusions. - The second law of motion says, that a body in motion is acted on by a force, in amount and direction the same as a body at rest. Now, in actuating the motion of a pendulum, it cannot be said that any motion is imparted at all, but only a direction given to a motion derived from gravity; now, if direction only is the thing imparted, the force for the motion existing before, it is this direction only that is independent and likely to remain unchangeable. Besides, I understand the nicest experiments have been made long ago, proving that a pendulum vibrates in exactly the same time in whatever azimuth it oscillates; and as the earth's rotation, if it could have any effect in deflecting the plane of a pendulum hanging free, would have an equivalent effect in retarding the motion of one which, for such experiments, must be constructed to move in a prescribed direction, we may presume no such effect belongs to the rotation. Still, on this last point, it must be conceded, that it is perhaps not known whether the times of a pendulum hanging quite free would be the same as one constrained to vibrate in one direction, as all those attached to common clocks are.

Another point may be just mentioned; it seems probable that any deflection of the plane produced by gravity during the motion of the point of suspension in a circle, would not be all one waythat what might be produced in one direction through the first quadrant

would be reversed in the second-what might be produced in the third would be reversed in the fourth, or vice versa. If these views raise the presumption that the direction of the plane may be invariable at all latitudes, or that at any rate the compendious method of the two