66-68] General Theory 67 



the lower limit in equation (134) now being the positive root of the equation 



F=Q ................................. (136). 



By differentiation of equation (134) we obtain 



x x ^ x 



which, in virtue of equation (136), is equivalent to 



while differentiation of equation (135) gives directly 



dX ................. ...(138). 



At the boundary, X = ; whence it is clear that F = F; at the boundary 

 and that all the differential coefficients of F are equal to those of F;. At 

 infinity X = oo ; whence it appears from equation (134) that F = at infinity. 

 It accordingly appears that equations (134) and (135) will give the true values 

 of the potentials provided 



V 2 F = ............................ (139), 



............................ (140). 



68. By further differentiation of equation (137), we have 



so that 



V^ =J%(X)VWX-^(X)2g T g ............... (141), 



while, for the simpler function F;, we have 



d\ .................... (142). 



Now suppose that F is such as to satisfy at all points of space the equation 



Suppose further that F is such that 



at infinity i.e. when X = oo . 



Then at infinity equation (143) reduces to 



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