PROFESSOR KELLAND ON THE THEORY OF WAVES. 119 



have 26=0 (12). This is satisfied by giving to n pairs of equal values positive 

 and negative ; and thus we get as the general solution of the equation, subject to 

 our conditions 



/= 2 b f e* 



the 2 embracing all the positive values of n. 



By equations (4) and (5) we obtain the values of F and F thus : 



F= - 2 b Vl^r? [e* nz + ' n 2 }{e K 

 F=-lbn { e '^_ e -"-]{ e a Vi=^ 

 Thence equation (10) gives 





at the surface. 



Also equation (11) gives 



g 



~ a ' 



at the surface. 



Equations (13) and (14) contain the solution of the problem. 



60. APPLICATION. The most simple type which a wave can have is that which 

 is expressed by one function only. In this case 2 may be omitted, and the value 

 of (? is independent of y, or is the same throughout the whole mass ; all our pro- 

 cesses are consequently applicable without any limitation to this case, and we 

 may regard our solution as a complete one. 



As an approximation, let us expand the exponentials contained in these 

 equations : by this means we get 



^MA<<t 



or j- ="V (z) nearly, 



and 

 Hence 



COB. If y =a n z m , which case includes the triangle, parabola, &c. we have 



