466 On ^Eolotropic Potential. 



is taken to define a function of the position of ]S T , viz F(N) • 

 then the means of/(P) and F(N) over the sphere are equal/ 

 In seolotropic potential each integral form has a factor 

 A p or 1-Xp 2 m the numerator ; the first is also modified by 

 a divisor u p (l,m,n), and the second bv a divisor ^/uJTmTri!') 

 (hnn) defining the direction of the normal N to each plane 

 section, (I m'ri) the direction of a radius OP. Let (Imn) 

 correspond to (to), {Vm>n<) to {&<}>>) in a polar system with 



i T!^ 18 '^ } et - x be the an 3* le betw een the planes OPZ 

 and OPN, NP being a quadrant. As above, the triple 

 Integra is represented by d X dco', but the divisor u p is different 

 lor each plane section of the cone about OP, and u v is a 

 function of x "hile the other part of the integrand is 

 independent of y. From the spherical triangle ZPN we 

 have (1) cos0 = sin^cosy, (2) sin % = sin d sin (<£' I £), 

 and (3) cos 6' cos 6 + sin & sin 6 cos (<// - A) = or by O ) 

 cos 0' cos y + sin 6 cos (<£' - fl = 0. With J = sin d cos ci, 

 m = sm sin <£, n = cos 0, these give 



Z = sin % sin <£' — cos ^ cos <£' cos <9', 



— m = sin % cos <£' + cos y, sin 0' cos 6", n = sin & cos X . 



Thus u p assumes the form a sin 2 y + /3 cos 2 y> 2 7 sin x cos y,, 



andJ o ^K^27r/v / ay S-7 2 . It will be found that 

 «/3 - 7 2 = PZ' 2 + Q m '2 + Rn / 2 + 2 p/ mV + 2Q , w7 , + 2RTm , 

 = wp(/', m\ n'). 



Thus the two modifications exactly correspond. On the 

 one view of potential the element is a plane area, which in 

 seolotropic potential is modified by a divisor depending on 

 the exposure of the plane in relation to the seolotropic axes 

 viz. v p (lmn) in the general case, l-(2//0 2 in the motional 

 case. On the other view the element of volume is divided bv 

 distance from the point attracted; in seolotropic potential 

 this is modified by a divisor which changes 



^(w-x'y to ^Mx=¥~,y-y\ z-7), 

 and in the motional case to 



v y (L-V)2(^^ / ) 2 + (2p(^-i 7 )?. 



We may characterize these two forms of the expression for 

 potential as the wave and the ray forms. 



