44 
Proceedings of the Royal Society of Edinburgh. [Sess. 
path is shorter than the reference path by OM + ON. Hence the amplitude 
of the ray traversing it is proportional to 
cos {gt + &(OM + ON)}, 
where Jc = 27 r/\ and ^ = 27 t/t, A and r denoting wave-length and period. 
Form similar terms for the other electrons, take the sum, and we obtain 
cos gt 2 cos &(OM + ON) - sin gt 2 sin &(OM + ON). 
The intensity of this wave varies as 
[2 cos k{ OM + ON)] 2 + [2 sin ft(OM + ON)] 2 ] 
n + 22[ cos &(OM a + ON a ) cos &(OM 5 + ON & ) + sin k( OM a + ON a ) sin k(OM b + ON*)] > (1) 
n + 22 cos k( OM a - OM b + ON fl - ON 6 ) j 
where the suffixes a and b refer to the a th and 6 th electrons in the atom, 
and the summation is taken over all the \n(n— 1) pairs of electrons in 
the atom. 
Let us consider the pair formed by the a th and 6 th electrons a little 
more closely. Let A and B denote their positions in space, and let L be the 
distance between them. Let ZB and BQ denote the directions of the 
incident and scattered light. Let BG bisect L ZBQ. Draw AM per- 
pendicular to BZ and AN perpendicular to BQ. Then the typical term 
under the summation sign becomes 
cos &(BM + BN) = cos &L (cos ZBA + cos ABQ). 
In fig. 3, Z, G, A, and Q represent the intersections of the lines BZ, BG, 
BA, and BQ with a sphere of unit radius with its centre at B. Then by 
a formula in spherical trigonometry 
cos ZBA = cos ZG cos GA + sin ZG sin G A cos ZGA 
and 
cos ABQ = cos GQ cos GA + sin GQ sin GA cos AGQ. 
But ZG = GQ = Jd. Hence, by adding 
cos ZBA + cos ABQ = 2 cos cos GA = 2 cos cos <£ 
