in reference to the Theory of Ocean Currents. 197 



for the depth of the penetration of the surface-impulse within 

 a fixed time. 



The general solution of the differential equation (1), with 

 the accessory conditions (2), (3), (4), was given by Poisson*. 

 It can be derived more elegantly according to the method 

 (probably originated by Dirichlet) which > in Riemann's Lec- 

 tures on Partial Differential Equations, p. 140, is applied to 

 the somewhat simpler problem when there is no free radiation, 

 but a given temperature at the surface. 



The function w may be composed of two functions u and v y 

 which satisfy the general differential equation (1) for w y and 

 fulfil the following conditions : — 



For^ = 0. For £' = /*. For£ = 0. 



- Yx +pU = °' U = °' U = ^)> 



- g^ +pv =i*£(0j v = o, 9 = o.. 



Their sum, u + v = w, obeys then the conditions (2), (3u),and 



If m denotes the infinite number of collectively real roots 

 of the transcendental equation 



m cos mh +p sin mh = Q, ..... (5) 



the demands for u are fulfilled by the following expression 

 (which is to be summed over all the roots m) : — 



•^^s^^rf^^-* 4 -^^ (6) 



Those for v are fulfilled by 



_ , , v f p(h — x) 9 ^ m cos mx + p sin mx 1 



The first member serves only for representing the function for 

 x~0, and vanishes as often as 0<x^h; for its second term 

 is nothing more than the development of the first according 

 to the sine of the argument (Ji^x) multiplied by the roots of 

 the transcendental equation (5), and consequently disappears 

 together with the first term whenever x > 0. For the applica- 

 tion to internal points this first member can therefore be 



* Journal de VEcole Poly technique > cah. xix. p. CO; ami Theorie Mathe- 

 matique de la Chaleur, p. 327. 



