668 



wilson: geometric potential 



of the charge, and I is the idemfactor, it is easy to show that 



0-0p= -2cU» / + £ +w 



u 



<P 



+ 1 



J" + 



R, 

 R 



2/ + U+- 



IR V R 



V + 21-c) 



] 



(l+21.c)+2/ , (l+l.c) 



If this is to vanish, / + <p/R = 0, that is 



A 



<p = 



R 



/ = 



3. As the potential has turned out homogeneous of degree 

 zero in w, we may write 



w u 



P = ~ ;— = - .— 

 l'W 1-u 



where u is any tangent to the curve 5. If we define dq by the 

 relation dr = u dq, where dr is the increment along the curve, 

 we have determined a parametric representation of the curve 

 so that u = dr/dq, which is analogous to w = dr/ds, but more 

 general in that it would be applicable to curves of zero length. 

 The equations for the derivatives would now become 



0u = 



1-u 



01 = /- 



hL 

 1-u 



0(l.u) = (01).u+(0u).l = u-~l+-l 



l,u 



1.11 



where 

 Then 



and 



c 1 = du/dq 



0v = 



0xp = 

 P= 0xp = 



uu 



(1-u)- (1-u) 3 



u-u , . 1-c 1 . lc 1 



lu + — — — lu - 



a-w) 1 



d-uy 



U-U , , 1-C 1 . IXC 1 



lxu + - - Ixu - 



(l;u) ; 

 u -u 



lxu + 



a-u)« (huy 



lxfl^uxc 1 )] 



(1-u) 3 " (1-u) 3 



The vector P, of the second sort, is the (geometric) field 9 set 



9 Wilson and Lewis, loc. cit., p. 460. 



