8 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



3. Let /be a quantity which may be regarded as a function of the rectangular co- 

 ordinates of a point of space, or simply, without the aid of co-ordinates, as having a given 

 value for each point of space. It will be supposed that / vanishes outside a certain portion T 

 of infinite space, and that within T it does not become infinite. It is required to determine a 

 function U by the conditions that it shall satisfy the partial differential equation 



VUmf. (4). 



at all points of infinite space, that it shall nowhere become infinite, and that it shall vanish at 

 an infinite distance. 



These conditions are precisely those which have to be satisfied by the potential of a finite 



/ 

 mass whose density is ; and we shall have accordingly, if be the point for which the 



47T 



value of U is required, and r be the radius vector of any element drawn from 0, 



tt. 



a —km"- (5) - 



In fact, it may be proved, just as in the theory of potentials, that the expression for U given by 

 (5) does really satisfy (4) and the given conditions; and consequently, if U + U' be the most 

 general solution, U' must satisfy the equation y V = at all points, must nowhere become 

 infinite, and must vanish at an infinite distance. But this being the case it is easy to prove 

 that V' cannot be different from zero. 



The solution will still hold good in certain cases when / is infinite at some points, or when 

 it is not confined to a finite space T, but only vanishes at an infinite distance. But such 

 instances may be regarded as limiting cases of the problem restricted as above, and therefore 

 need not be supposed to be excluded by those restrictions. 



4. Let U be a quantity depending upon the time t, as well as upon the position of the 

 point of space to which it relates, and satisfying the partial differential equation 



d*U 



d -=a=y£7. ........ (6). 



It is required to determine U by the above equation and the conditions that when t = 0, 



U and — — shall have finite values given arbitrarily within a finite space T, and shall vanish 



outside T. 



Let O be the point for which the value of U is sought, r the radius vector of any element 



drawn from O; f(r), F(r) the initial values of V, — . By this notation it is not meant that 



these values are functions of r alone, for they will depend likewise upon the two angles which 

 determine the direction of r; but there will be no occasion to express analytically their 

 dependance on those angles. The solution of the problem is 



U=~ffF(at)d ff + ^-~tfff(at)da (7). 



47T 47T at 



See a memoir by Poisson Mem. de fAcadimie, Tom. HI. p. 130, or Gregory's Examples, 

 p. 499. 



