Mk. cayley, on the theory of determinants. 



81 



Also 



FU = - 



A.SC + K w + . . , HX + 'B 31 + . 



(.-il), 



k'^ + s'r; 

 which may be transformed into 



FU = 



AX + A X . . . , BX + S X 

 p ) c 



R^ + s>; . . . , n'^ + s'>; . . . , 

 a , a 



b , b' 



(52), 



(for shortness, I omit the demonstration of this equation). 

 And similarly, 



1U = 



Tix + s. X , . . , s ,r + s a; . 



p 5 o" 



P 5 O" 



A^ + B>;. . , a'^ + s't?. . 



a , a 



b , b' 



(53), 



where it is obvious that if the sum 2 contain fewer than (m - 1) terms, FU = 0, Y(7 = 0. 



The equations (52), (53) express the theorem, that whenever KU = 0, the functions FU, 'JU 

 are each of them the product of two determinants. 



If next 



u,= u +u. 



Taking g = — \, in the formulae 



K.(K(U+U)FU - KUF{U+U)) = K .{K (U+U) 1U - KU1 (U+U)) 



= {KUy-'.{K(U+U))"-'.KU. 

 Or observing the equation (50), 

 K.{K{U+U)FU -KU.F{U+U)) ^ K.{K {U+U) 1U - KU1 (U+U)) = 0. 



Hence F .{K {U+U) 1U - KU1 (U+U)) = 1 . {K (U+U) FU - A'UF(U+U)) are each 

 of them the product of two determinants. But this result admits of a further reduction. We have 



F . {K (U+U) 1U - KU1 {U+U)) = 1 . {K (U+U) FU - KU. F (U+U)) (58) 



(56). 



.... 57. 



= - J/" (KU)'-' .{K(U+ U))"- 



a^x + a' x . . , /3,'F + (^l x' . . 

 a^ + fit] . . a^ — a, /3^ — /3 



Substituting a^ = a + '2pa, &c. .. also observing that if the second line be multiplied by x, 

 the third by x', . . and the sum subtracted from the first line, the value of the determinant is 

 not altered, and that the effect of this is simply to change a,, af . . into a, a'., in the first line, 

 and introduce into the corner place a quantity - U, which in the expansion of the determinant 

 is multiplied by zero. This may be written in the form 



-jr{KUY-^{K{u+ u))" 



Vol. VIII. Paiit I. 



ax + ax + . . , /3a.' + (i' x . 

 "? + ftrj "S-pa , '2<7a 



a'^ + (i' ri 'Zpa , "Zira 



(59), 



