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LXIV. On the Mathematical Principles of Laplace's Theory of 

 Tides. By Professor Challis, M.A., F.R.S., F.R.A.S* 



TN three articles, contained in the Numbers of the Philosophical 

 Magazine for January, April, and June, 1870, I have dis- 

 cussed the problem of the tides of an unbounded ocean of small uni- 

 form depth, including in the discussion several tentative solutions 

 of the problem, one of which I finally considered to be essentially 

 correct. In the course of endeavours to make the logic of that 

 solution as exact as possible, I arrived at the proof of a general 

 proposition relating to oceanic tides, which, at the same time 

 that it answers the purpose intended, applies also in an impor- 

 tant manner to Laplace's mathematical theory of tides. The 

 proposition may be thus enunciated : — Whatever be the circum- 

 stances of the tides, it is necessary to suppose that udx ~f- vdy + wdz 

 is an exact differential, the velocities u, v } w being resolved parts, 

 parallel to rectangular axes fixed relatively to the earth, of the 

 tidal motion at the point whose coordinates are x, y, z. The 

 proof above referred to is that which I now proceed to give. 



Since, on the principle of the supposition of small periodic 

 motions, the tides of the ocean, whatever be the forms of its 

 bottom and boundaries, are the sum of the tides which the sun 

 and moon would produce separately, we may calculate the effect 

 of the attraction of each luminary by itself. Supposing at first 

 the attracting body to be the moon, let the earth's centre, con- 

 ceived to be fixed in space, be the origin of rectangular coordi- 

 nates, its axis coincide with the axis of z, and the plane of its 

 equator coincide with that of xy ; and let the axes of x and y 

 have given directions in space. Then if x v y lt z 1 be the coor- 

 dinates of the moon's centre referred to these axes at any time 

 t reckoned from a given epoch, the values of these coordinates 

 are obtainable from the Lunar Theory as functions of t. Let 

 the axes of x and y be now changed to others in the same plane, 

 but fixed with reference to the rotating earth, and let the positive 

 direction of z be from the plane of the equator northwards, that 

 of y from the earth's centre to the meridian of Greenwich, and 

 that of x from the plane of that meridian eastward. Then, sup- 

 posing the new coordinates to be x 1 and y' } it will be found (the 

 earth's rotation being eastward) that 



x l = x l cos (cot + ci) —y x sin (cot + et), 

 y' = x ] sin (cot + a.) +y x cos ((ot + ct), 



o) being the rate of rotation, and a an arbitrary constant such 

 that if t x be the epoch of coincidence of the two sets of coordi- 



* Communicated by the Author. 



