334 HITCHCOCK. 



by designating the omitted vector. The other denominators may be 

 transformed in a simihir manner. 

 Consider now the identity 



F/3iCi + F132C2 + • • • + F^yCj = (13) 



where FjSi is any quadratic function of /3i, F^o is the same function 

 of /Si, etc. 3 If we let FX = (X|3im) (Xftv) we shall have i^/3i= and 

 the identity becomes 



(ft^iM) (Mip)C, + i^sM i^zMCz + • ■ • + (Miij) (^iMCi = (14) 

 which by the notation already employed may be written 



• • • + (21m) (21^)^2 + (31m) (31v)C3 + • • • (71m) i71v)C, = (15i) 

 and similarly by letting F\= (X/Ssm) (X|32J'), 



(12m) (12z^)Ci + • • • + (32m) (32v)C3 + • • • (72m) (72^)^7 = (162) 



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



(17m) (17i')Ci + (27m) i27u)C2 -\ + (67m) (67i')C6 + • • • (167) 



If we multiply these equations respectively by Ci, C2- ■ -Ct 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) (121^) = (21m) (21^). 

 Cancelling the factor 2 we thus have the new identity 



S(12m) {12u)Cia2 = (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 Cj to vanish. The remaining ten 

 terms may be arranged as follows, 



(561 (567)C5C6 + (641) (647)C6C4 + (451) (457)C4C5 

 + C2[(241) (247)C4 + (251) (257)C'5 + (261) {267)Ce] 

 + C3[(341) (347)C4 + (351) (357)C5 + (361) (367)C6] 



+ (231) (237)0-20 s = (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 /J? 

 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. 



