PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 9 



5. Let He a function which has given finite values within a finite portion of space, 

 and vanishes elsewhere ; and let it be required to determine three functions £, »/, £ by the 

 conditions 



i= 0; (8) - 



(9). 



The functions £, t], £ are further supposed not to become infinite, and to vanish at an infinite 

 distance. To save repetition, it will here be remarked, once for all, that the same supposition 

 will be made in similar cases. 



By virtue of equations (8), £ dx + r/dy + t£dz is an exact differential d\}/, and (9) gives 

 y \|/ = $. Hence we have by the formula (5) 



*=-hlII~r dV - CM), 



and \// being known, £, t], £ will be obtained by mere differentiation. To differentiate \j/ with 

 respect to x, it will be sufficient to differentiate § under the integral sign. For draw 00' 

 parallel to the axis of x, and equal to A*, let P, P' be two points similarly situated with 

 respect to 0, O', respectively, and consider the part of \^ and that of yjs + A \i/ due to equal 

 elements of volume d V situated at P, P' respectively. For these two elements r has the same 

 value, since OP = O'P", and in passing from the first to the second S is changed into 3 + AS, 



and therefore the increment of ^ is simply d V. To get the complete increment of \^ 



we have only to perform the triple integration, an integration which is always real, even 

 though r vanishes in the denominator, as may be readily seen on passing momentarily to polar 

 co-ordinates. Dividing now by Ax and passing to the limit, we get 



dS dV 



e-tt— J- m*l*i dt). 



* dx 4tt JJJ dx r 



By employing temporarily rectangular co-ordinates in the triple integration, integrating 

 by parts with respect to x, and observing that the quantity free from the integral sign vanishes 

 at the limits, we get 



^-hJIS^ C0 ^ rx)dv - (12) ' 



as might have been readily proved from (10), by referring Oto a fixed origin, and then differen- 

 tiating with respect to x. The expressions for rj and £ may be written down from symmetry. 



6. Let •ar', w ", nr"' be three functions which have given finite values throughout a finite 



space and vanish elsewhere ; it is required to determine three other functions £, v, t by the 



conditions 



dt dt] , dP dt „ dt] dP „, ,_ 



-T -^~ = 2 *r, -A" 7^ = 27«r", - r L--±=2 W "\ (13). 



dy dz dz dx dx dy 



dP dn dT , . 



dx dy dz 

 Vol. IX. Part I. 2 



