83 Mr. CAYLEY, on THE THEORY OF DETERMINANTS, 



which may be reduced to 



jr.(,s-uy-'. {K{U+ U))"-' X 



(60), 



aw + a 00 + . . , (iw + ^ CO + . . 



i^ + /3>;.., a'J + ^'v- ■• 



If each of these determinants were multiplied by the quantity (JTf/)""', expressed under the 

 two forms 



J, B .. 

 J', B'. 



They would become respectively 



KU 



Ap + B<r , A'p + B'fj 



A, A'.. 



B, B' 



(fil). 



KU. 



Aa + A' a . . , Ba + B'a! . 



(62). 



So that finally 

 F.{K{U+ U)'JU-KU.'J .iU+ U))=1.{K{U+ U) FU - KU . F {U + f7)) = 



{K(U+U)\-'-x 

 -^^ ■[ KU I 



Ap+B(T . . , A'p + B'a . . , 



Aa + A'a . . , Ba + B'a . . . 



... (63). 



The second side of which may be written under the forms 

 (K{U+U)Y-'-' AX+AW'.. , d.i+bV.. , .. 



A . (Ap + B<x ..) + a'. (A'p + B'a . •)• ■ , b. (Ap+ Ba ..) + n'.(A'p + B'a . .) .. 



j K(U+U)y 

 [ KU ) 



R^ + S,,.. , R'^ + Sn-. 



R.(Aa + A'a'..) + S . (Ba + B'a . .) .. , R'.(Aa + A'a . .) + S'.(Ba + B'a . .) 



...(64). 



And 

 V KU 



j K(U+U) Y 



R.v+R'w'.. , S.F+S'w 



R.(Ap + Ba . . )+R'.(A'p+R'a.. ).., S. (Ap + Ba . ■) + S'. (A'p + B'a ..)., 



Al + Br,.. , A'^ + B'n-. 



A.(Aa + A'a'..) + B .(Ba + B'a ..).., A'.(Aa + A'a'..) + B'.(Ba + B'a'..) 



And again, by the equations (52) (53), in the new forms 



(t^)"'- '^- ^H(^P + Ba..)(A^+^„..) + (A'p + B'a..) (A'f + b',, + ..)..] X 

 [(Aa + A'a' . . ) (u.c + n'.v' .) + (Ba + B'a . . ) (sa- + sV . . ) . . "i} 



(^YW^T' ^^ ■ ^^^"^f + ^<^ ••)("?+ ^-7 ■■) + (^y +B'a..) (R'e + s'„ + ..)..] X 

 \_(Aa + A'a' . . ) (a.v + aV . . ) + (Ba + B'n . . ) (B.r + n't;' • . ) . ■ ]j. 



(65). 



(66), 



(67). 



Comparing these latter forms with the two equivalent quantities forming the first side of (53), 

 and observing (33), (34). It would appear at first sight that 



