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LYII. On the Exact Form and Motion of Waves at and near the 

 Surface of Deep Water. By William Joiin Macquorn 

 Kankine, C.E., LL.D., F.R.SS.L. $ E. $c* 



THE following is a summary of the nature and results of a 

 mathematical investigation, 'the details of which have been 

 communicated to the Royal Society. 



The investigations of the Astronomer Eoyal and of Mr. Stokes 

 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 investiga- 

 tions, when applied to waves in which the displacements are 

 considerable as compared with the length of wave, are only 

 approximate. 



In the present paper the author proves that one of those 

 results — viz. that in very deep water the particles move with a 

 uniform angular velocity in vertical circles, whose radii diminish 

 in geometrical progression with increased depth, and conse- 

 quently that surfaces of continuity and equal pressure, including 

 the upper surface, are troehoidal — is exact for all possible dis- 

 placements, how great soever. 



The troehoidal form of waves was first explicitly described by 

 Mr. Scott Russell; but no demonstration of its exactly fulfilling 

 the cinematical and dynamical conditions of the question has 

 yet been published, so far as the author knows. 



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

 the author stated that the theory of rolling waves might be de- 

 duced 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 inves- 

 tigation on which that statement was founded. 



Having communicated some of the leading principles of that 

 investigation to Mr. William Froude in April 1862, the author 

 was informed by that gentleman that he had arrived indepen- 

 dently at similar results by a similar process, although he had 

 not published them. The introduction of Prop. II. between 

 Props. I. and III. is due to a suggestion by Mr. Froude. 



The following is a summary of the leading results demon- 

 strated in the paper. 



Proposition I. — In a mass of gravitating liquid whose particles 

 revolve uniformly in vertical circles, a wavy surface of troehoidal 

 profile fulfils the conditions of uniformity of pressure, — such 



* Communicated by the Author, being an abstract of an investigation 

 read before the British Association, 1862, Section A. 



