84 Mk. CAYLEY, on THE THEORY OF DETERMINANTS. 



To find the values of A, B, &c... Corresponding to U+U, we must write 



A = m'/3 + n'v' + — •• (7-t), 



= M ^ + N 7 + 



where 



m"= ± 



" S" 

 7)0.. 



7 ' 



in '\tn 



7.0 . . 

 y V 



, n'=± 



(75), 



Pi /// tl 



^"" e" 



The order of each of these determinants being n - 2, and the upper or lower signs being 

 used according as w — 1 is odd or even, i. e. as n is even or odd. Hence 



A^==A-\^ m'.o-o' + n'. to' + (76). 



m". era" + N".Ta"+ .. 

 And therefore 



K^A - kAi = A'^fja + AB . era + AC .ra + (77). 



AA'pa + {AB' - KM')crn'+ {AC - K^')Ta' + . . 

 AA"pa"+ {AB" - KM")aa" + {AC- /cn")x«" + 



The additional quantities C, r having been introduced for greater clearness. Now the 

 equations 



AB' - kM' = AB, AC - kN' = A'C, 

 AB"- kM" = A"B, AC- kN" = A"C, 



(78), 



written under the form 



AB' -A B = kA/', AC -A'C = kN', (79), 



AB"-A"B=kM", AC"-A"C=kN" 



are particular cases of the equation (13), and are therefore identically true. Hence, substituting 



in (77), 



HiA - kA, = A' pa + ABaa + ACra ... + (80), 



AA'pa + A' Baa + A'Ctu' ... + 



A"Apa" + A"Baa" + A"Cra" ... + 



= {pA + aB + ... ) {Aa + A'a + ... ). 

 Forming in a similar manner, the combinations k^B - kB^ ... k^A' - kA^', k^B - k B,', ... , mul- 

 tiplying by the products of the different quantities Ax + A'a; ... , Biv + B'x ... , ... 7?^ + St] + ... , 

 fi'5 + -S"^, ... and adding so as to form the function K {U + 17) . FU - KV . F {U +U), we 

 obtain the required formula, viz. that the value of this quantity is 



[{pA + aB + ..){r^ + s,i + ..) + {A'p + B'a..) (u'^+ s' ,,..) + ..] X (81), 



[{Aa + A'a' . . ) {a.v + aV . . ) + {Ba + B'd . . ) (nj' + uV +.) + -] 

 with this theorem, I conclude the present section, — noticing only, as a problem worthy of in- 

 vestigation, the discovery of the forms of the second sides of the equations (()8), (ft)), in the 

 case of 2 containing more than a single term. 



