334 HITCHCOCK. 



by designating the omitted vector. The other denominators may be 

 transformed in a similar manner. 

 Consider now the identity 



MCi + F^2C2 + • • • + F/37C7 = (13) 



where F^i is any quadratic function of j3i, Fj32 is the same function 

 of /32, etc.3 If we let F\ = {XM O^M we shall have F^i= and 

 the identity becomes 



(^,M Wip)C2 + i^sM m,v)C, + • • • + (^7/3im) {^l^lv)C^ = (14) 

 which by the notation already employed may be written 



• • • + (21/x) {2lv)C2 + (31m) (31v)C3 + • • • (71m) (71j')C7 = (15i) 

 and similarly by letting F\— (\^2ij) (X/^oj'), 



(12m) (121^)^1 + • • • + (32m) mu)C3 + • ■ • (72m) i72p)C-j = (ISs) 



Proceeding thus we obtain a set of seven equations of which the last is 



(17m) (17^)Ci + (27m) {27u)C2 + • • • + (67m) {C>7u)C, + • • • (15:) 



If we multiply these equations respectively by Ci, C2- • -C? and add, 

 we note that each term of the sum is of the form (12m) (12?^) C1C2 and 

 that each such term occurs twice, since (12m) (12i') = (21m) (21^). 

 Cancelling the factor 2 we thus have the new identity 



S(12m) (12j^)CA = (16) 



where the left side contains as many terms as pairs can be chosen 

 from the numbers one to seven, that is 21 terms. 



So far M and v are any vectors whatever. Now let m = /3i and v = ^7 

 causing all terms containing Ci or C7 to vanish. The remaining ten 

 terms may be arranged as follows, 



(561 (567)C5C6 + (641) {m)C,Ci + (451) (457)C4C5 

 + C2[(241) (247)C4 + (251) {257)C, + (261) (267)C6] 

 + C3[(341) (347)C4 + (351) (357)C, + (361) (367)C6] 



+ (231) (237)C2(73 = (17) 



Returning to (13) and putting F\ = (2X1) (2X7) we have, since the 

 first, second, and seventh terms vanish, 



3 This identity may be proved by noting that the left side is quadratic in 187 

 and vanishes when /S? coincides with any one of the other six vectors, hence 

 vanishes identically. For a more detailed consideration of the C's see "An 

 identity connecting seven vectors," Proc. Royal Soc. Edinburgh, 40, Part 

 II (No. 14), June 1920. 



