82 liErouT— 1881. 



The principle may be compared with that of Kewton's principle of 

 dynamical similitude. An example illustrating Helmholtz' results is 

 given by Rayleigh' on the analogy between the two dimensional vibra- 

 tions of air in a cylindrical tube of any section, and the liquid Avavea 

 contained in a vertical cylindrical vessel of the same section. 



/. Waves hi Liquids. 



The subject of waves was one which received much attention amongst 

 English mathematicians about the period between 1840 and 1850, and the 

 labours of Green, Kelland, Barnshaw, and Airy have been noticed by 

 Professor Stokes in the last report to the Association. Almost imme- 

 diately after this report he himself read a paper before the Cambridge 

 Philosophical Society 'On the Theory of Oscillatory Waves,' ^ in which 

 was investigated the form and properties of waves which are propagated 

 tvithout change of form and irrotationally. It appeared that with these 

 conditions^ given, to a given velocity of propagation corresponded one 

 definite form ; when the height is small compared with the wave-length, 

 the wave-form is the curve of sines ; but if the height is comparable with 

 the wave-length this is not the case, but the crests of the waves are 

 steeper than the hollows, and this difference becomes more prominent 

 the shallower the fluid is. A curious result of the analysis is that the 

 fluid particles, in addition to a motion of oscillation have also a small one 

 of translation, which depends on the square of the ratio of the height to 

 wave-length, a result which Rayleigh'* has shown to depend directly ou 

 the absence of molecular rotation of the wave. In this paper Stokes 

 carried the approximation to a second order for finite depths, and to a 

 third order when the depth of fluid is infinite. In the reprint of his 

 papers''' (1880) he adds a supplement to the above in which a totally 

 different method is employed. Instead of taking the rectangular co- 

 ordinates of a particle as independent variables and expressing the 

 velocity potential thereby as usual, the velocity potential and stream 

 functions are taken as independents and the co-ordinates of a point 

 sought in terms of them. This so much simplifies the calculation that 

 it is an easy matter to press the approximation to the fifth oi'der for 

 infinitely deep fluids, and to the third order for those of finite depth. In 

 infinitely deep fluids a wave-form of length 27r/wi and height 2a + ^m^ a^ 

 the velocity of propagation (c) is given by'' 



c2 = £ (l + vi^a- + fwV) 

 m 



> ' On Waves;,P7iil. Miff. (.5) i. p. 275. 



- Trans. Cawb. Phil. Sue. viii. p. 441. Also reprint, vol. i. p. 197. 



' It will be seen below that another conditiou is implied, viz., the finitenesa of 

 wave-length. 



■' ' On AVaves,' Phil. Mar/. (5) i. p. 270. 



5 Mathematical and Physical Pajjcrs, G. G. Stokea. Vol. I. Cambridge 

 Universit.y Press. 



" li h = height of crest above the trough and A = wave-length 





jit OS 



TT-h' 6rr .T , tt'//-* Sir x 



+ VfT cos — j; COS • + 



"' A- A ■- A' A 



