394 HITCHCOCK. 



form of (45a3), with two similar determinants for (SGas) and (64a3); 

 we have 



^23(123) (456) = (31a3) (456), by (60), 



= (314) (56a3)+(315) (64a3)+(316) (45a3), identically, 



whence, using (20) as above indicated, the last expression is the same 

 as the determinant 



A;4(314), (314) (124), (124) (234) 

 hlsi5), (315) (125), (125) (235) 

 A;6(316), (316) (126), (126) (236) 



If we expand by the elements of the third column we find 



(69) 



yl 23(123) (456) = (124) (234) (315) (316) [hil26)-h{125)]-{- ... + .. .; 



(70) 



but, by (66), k^ = — /S/CjSa and k^ = — Sk06, giving 



k5{l2&)-k6{125) = - 8^/35(126) + S/c/36(125) 



= + SK[0,S8i^2^e - /36S/3i|82i85] 

 = S/cF(F/3i/32F^6/35), identically, 



= S/c[/3iS/32/36^5 - ^2S^u86i86] 



= SKi3i(256) - S/c/32(156); (71) 



the second and the third terms on the right of (70) may be similarly 

 transformed, being obtained from the first term by advancing the 

 numbers 4, 5, 6. If we collect the coefficients of Sk^i we therefore 

 have 



(124) (234) (315) (316) (256) + (125) (235) (316) (314) (264) 



+ (126) (236) (314) (315) (245). 



In the first of these three terms, make the identical transformations 

 (234) (315) = (123) (345) + (235) (314) 



and in the third term, 



(236) (315) = (123) (365) + (235) (316). 



If the resulting five terms are grouped into those with the factor (123) 

 and those without it we have 



