r).'2 Royal Society : — 



several objects which, though unexamined, appeared to be certainly 

 worthy of notice ; and these are very properly distinguished by 

 being placed between brackets. The article on the moon deserves 

 particular notice. It comprises just a fifth part of the whole work, 

 and is an excellent sketch of " selenography." It is accompanied 

 by an elaborate map, slightly exceeding 12 inches in diameter, exhi- 

 biting the chief objects on the moon's surface that common tele- 

 scopes can be expected to reach. The writer has given, by way of 

 introduction, a few remarks on the telescope, and some rules for using 

 it, which cannot fail of being serviceable to beginners. 



XII. Proceedings of Learned Societies. 



ROYAL SOCIETY. 

 [Continued from vol. xxxv. p. 465.] 

 May 7, 1868. — Dr. W. B. Carpenter, Vice-President, in the Chair. 

 rpHE following communication was read : — 



■ "On Waves in Liquids." By W. J. Macquorn Rankine, C.E., 

 LL.D., F.R.S. 



(1) Object of this Paper. — It has long been known that in a 

 uniform canal filled with liquid, the speed of advance of a wave in which 

 the horizontal component of the disturbance is uniform from surface 

 to bottom is equal to the velocity acquired by a heavy body in falling 

 through half the depth of the canal. But, so far as I know, it has 

 not hitherto been pointed out that a similar law exists for waves 

 transmitting a disturbance of any possible kind in a liquid of limited 

 or unlimited depth, provided only that the upper surface of the 

 liquid is a surface of uniform pressure. The object of this paper is 

 to demonstrate that law, and to show some of its applications. 



(2) Velocity of Advance defined. — Throughout this investigation 

 the velocity of advance of a wave will be defined to be the mean be- 

 tween the velocities with which the shape of the wave advances, rela- 

 tively to a surface-particle at the crest and to a surface-particle in 

 the trough respectively. In ordinary rolling waves the velocities of 

 particles in those two positions are equal and contrary, so that the 

 speed of advance as above defined is equal to the speed of advance 

 of the wave relatively to the earth. A wave of translation in which 

 the velocities of particles at the crest and hollow are not equal and 

 contrary, may be regarded as produced by compounding the motion 

 of a rolling wave with that of a current whose velocity is half the 

 difference of the velocities of those particles. 



(3) Relation between height of wave and horizontal disturbance at 

 the surface. — The following relation between the height of a wave and 

 the horizontal disturbance of the surface-particles has already been 

 proved and made use of by various authors ; and it is demonstrated 

 here for convenience only. Let -riband — u x be the velocities of a 

 surface- particle at the crest and trough of a wave respectively. Let a 

 be the velocitv of advance of the wave as defined in article 2. Con- 



