94 Prof. J. W. Nicholson on Atomic Structure 



If the line CB is taken as an axis of x in a Cartesian system 

 with C as origin, the equations of AL, BM become 



_&sinCQ-CL) 

 V fcoosC-CL ' 



_ CMsinC(a?-fl) 



y CM cos C- a ' 



and the coordinate .2? of the point becomes on solution, 



gCL.(CM-6)-ft . CMcosC(a-CL) 

 CL.CM-^ 



If r 2 = r 3 , so that two of! the electrons are in a ring, x is ^a, 

 and this must be true approximately if the deviations of the 

 two electrons from their ring exist, but are not large. 



This condition reduces to 



a"b + a 2 GL.CM-2a 2 b.CL = CK(a-Gh)(b 2 -j-a 2 -c 2 ); 



and substituting for CL and CM, the law of force f(r) must 

 be consistent with 



« { V(e) + cf(b) } { cf(a) + af(c) } = a%f{c) + 2abcf(a)/(c) 

 + c(b* + a*-c>)f(b)f(c) 



for all values of a, b, and c. No law which is a power of r 

 will ordinarily serve, for writing f(r)=\r n , we have 



a{bc n + cb n ) {ca n + ac n ) = a 2 bc 2n + 2abc . a n c n + c(^ 2 + « 2 — c 2 )b n c n , 



or on reduction, 



(52_ c 2) c »-i 6 »-i =fl »+i(^-i-c n - 1 ). 



The case b = c is a solution for any such law. If b is not 

 equal to c, any case of n negative (and the case w = 0) makes 

 one side of the equation essentially positive and the other 

 essentially negative, so that it cannot be satisfied. The case 

 n = l compels b and c to be equal. n = 2 gives a= [5c(6 + c)]i, 

 unless b = c, and in fact, laws of direct distance beyond the 

 first are formally possible. But they cannot be permissible 

 from physical considerations, for the repulsion between 

 electrons cannot be thought of as increasing with their 

 distance apart, when that distance is comparable with the 

 atomic radius, as in Bohr's rings of electrons. 



If the distance apart were comparable with the radius 

 of an electron, of order 10 ~ 13 , such a possibility might be 

 admitted, and in a subsequent paper it will be found to lead 

 to valuable conclusions. But here we are not concerned with 

 possible nuclear structure, of which this is a question, but 



