639 



?<;,, =f — {1 — 5 co.s' (9, — 5 cos^ e., — 1 5 m»'' 6, cos' 6^ A- 



+ 2 (4 cos &^ cos d^ + sin 6^ sin 6^ cos (fY\, .... (5) 



if fij represents the iiioiDent of the quadruplet. 

 We introduce: 



., f*/ 



■> ^5 



(«i) 



O" 



tlien r ^ the potential energy, when two molecules are touching 

 each other, the axes of the quadruplets being at right angles to 

 each other and to the line joining the centres. 



We put further 



Wr= 1 — 5 cos' (9i — 5 COS' 6», — 15 cos' 6, cos' (9, + 2 (4 cos 6, cos 0^ + 

 + sin (9, sin 6^ cos (f)', 



or: 

 if 



so that 



'ƒ' = A-irB cos <p + C cos 2r/' (7) 



^ = 2 (1 — 3 cos' Ü,) (1—3 cos' 6^ 1 



B ^\Q sin 6^ cos 6^ sin 6^ cos 8^ .... (8) 



C = sin' 8^ sin' 0.^, 1 



0^ 



Developing e~'"'i'i — 1 into a series of ascending powers of hui,\, 

 and integrating in (3) according to r, we obtain'): 



P'=iö» "^ ( — 1)» T"— (/"')" lil V^" sin 8, sin 8^(18, 'W^dq . (9) 



(I 



If for the sake of brevity we write [•/""] for the integral in (9) 

 and correspondingly : 



71 71 



lAi'B''iC'] = i I Ai'B-^'iC'- sin 8, sin 8, d8, d8A 



' 



'2ff] =. I cos'' ([ cos^ 2(f dff, \ 



0") 



Icvs'' ff' cos" 



observing that [cos'^'-ir/-] =: 0, [co«'''-i2r/'] ^ 0, [cos-i-—^rfcos'"'f\^() 

 I and m being positive integers, we find : 



') The quantities n, p, q. r, s, vvhicii we introduce temporarily in this g have a 

 meaning dilTerent from Uiat in the other part of this paper and in Supph No. 24. 



