1862.] 275 



II. The force per unit of area is given in magnitude and direction 

 at each point of each surface. 



The formulas are applied to determine the deformation produced 

 in the earth by the tide-generating forces of the moon and sun, on 

 certain definite hypotheses as to the rigidity of the earth. Thus the 

 theoretical results used in a previous communication by the same 

 author, "On the Rigidity of the Earth" (Proceedings of the Royal 

 Society, May 15, 1862), are proved. 



II. " On the Exact Form and Motion of Waves at and near the 

 Surface of Deep Water." By Professor W. J. MACQUORN 

 RANKINE, C.E., F.R.S. &c. Received September 27, 1862. 

 (Abstract.) 



The investigations of the Astronomer Royal and of other mathe- 

 maticians on the question of straight-crested parallel waves in a liquid, 

 are based on the supposition that the displacements of the particles 

 are small compared with the length of a wave. Hence it has been 

 very generally inferred that the results of those investigations are 

 approximate only, when applied to waves in which the displacements, 

 as compared with the length of a wave, are considerable. 



In the present paper, the author proves that one of those results, 

 viz., that in very deep water the particles move with a uniform 

 velocity in vertical circles whose radii diminish in geometrical pro- 

 gression with increased depth, and consequently that surfaces of 

 equal pressure, including the upper surface, are trochoidal, is exact 

 for all displacements, how great soever. 



The trochoidal form of waves was first explicitly described by Mr. 

 Scott Russell ; but no demonstration of its exactly fulfilling the cine- 

 'matical and dynamical conditions of the question has yet been pub- 

 lished. 



In 'A Manual of Applied Mechanics' (first published in 1858), 

 the author stated that the theory of rolling waves might be deduced 

 from that of the positions assumed by the surface of a mass of water 

 revolving in a vertical plane about a horizontal axis; but as the 

 theory of such waves was foreign to the subject of the book, he 

 deferred until now the publication of the investigation on which that 

 statement was founded. 



