{36 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



or if *~ 4- (*-*)*=0. 



Consequently the particles of the fluid which lie above the equilateral hyperbola 

 defined by this equation will commence to move upwards : those which lie lelow 

 it will commence to move downwards ; and those which lie along it will at the 

 first instant be at rest. 



/ 2 _o2\ _, b _ x 2 



4. -.= 9 -0^7 where r, r' are the distances of the point 



ay, 7T r . r * 



whose motion is determined from the two extremities of the summit of the ori- 

 ginally elevated fluid. 



Combining these remarks with the conclusions previously arrived at, Art. 70, 

 we conclude that no part of the fluid, except that which is in the immediate vici- 

 nity of the disturbed particles, commences to move with a wave motion ; but that 

 it gradually swells so as to diminish the quantity of fluid which actually consti- 

 tutes the volume of a wave. 



These conclusions are of sufficient importance to warrant us in bestowing a 

 little more attention on this part of the subject ; especially as M. POISSON'S result, 

 although in some points the same as our own, in others differs materially from it. 

 For instance, M. POISSON'S line of no vertical velocity is a straight line. 



73. Let us then determine the value of </> corresponding to very small values 

 oft: 



a 

 We have <b = I 



IT J o 



But r ^f^ 6 



/ o 



bg t ( _i b y _i by 



{ TT tan tan 



I- 1+' 



a a 



a 



