80 



Mb. CAYLEY, on THE THEORY OF DETERMINANTS. 



Let U, a, 3 , ... J , B, &c. . . be analogous to U, a, /3..., A., B, &c. . . and consider 



the equation 



K. {KU,FU + g. KU. FU,) (40). 



= I K,A+gKA^, K,B + gKB^. . 

 K^A'+gKA^' , K^B + gKB\ 



Multiply the two sides by the two sides of the equation (2), the second side becomes, after 

 reduction, 



K,K+gK.{A^a + B^^ + ..), gK.(,A;a+B;f3 +..) (41). 



gK.{Aa'+B^li'+..), K,K+gK.(A;a'+B;i3' + ..) 



Multiplying by the two sides of the analogous equation 



a , a . . 

 A> fir- 



and reducing, the second side becomes 



KK^ . {a, + go), KK, . (/3, + g(i) . . 

 KK, . (a/ + ga), KK, . (^/ + gfi') 



(42). 



(43). 



= ^\K;.K(U,+gU) (44). 



whence 



K. (KU,. FU + gKUFU) = {KU)"-'. iKUy-'.K{U, + gU). 



and similarly 



K. iKU,1U + gKU1U,) = (KUy-\(KUJ-\K.(,U, + gU). 

 In a similar manner is the following equation to be demonstrated, 

 1 . {KU,FU + gKUFU;) = F {KU;iU ^ gKU^U;) = . . 



(47). 



(45). 

 (46). 



-jf-iKuy-'iKUx-'x 



a,x + a'x . . , (ix + /3/ir'. . 

 aj + ^>).. a, + ga , /3^ + gfi 

 a'^ + fi'v «; +ga , /3; +g-/3' 



Suppose 



f7= 2 (p^ + art] + ...) (ax + a'x' + ...) 

 This expression being the abbreviation of 



U = (p^ + cr>; + . .) (oa? + a'x' + . .) + 



(/"/? + °,1 + ■ •) («,* + a/'»'+ • ■) + 



+ 



(48). 

 • (49)- 



[(w - 1) lines, or a smaller number]. 



KU = 



Sap, 'S.aa . ■ 

 Sa p, 2a'cr . . 



(50), 



which follows from the equation ( O )■ 



Conversely, whenever KU =0, U is of the above form. 



