MinorDeterminantsofLinearlijEquivalentQuadraticFiinctions. 299 



generall)^ by a^.a^, bj)^ which enter into U and V. Our object 

 is to be able to express the minor determinant 



Li/j 6/2 •• • ^'.- J ' 

 in which the one group of distinct numbers, A:,, k^. . . k^ may 

 either differ wholly from, or agree wholly or in part with the 

 other group of distinct numbers l^,l^...l, under the fomi of 





The particular value of Q corresponding to each double group, 



/, may be denoted by Q ^ ^ '• ; so that our 



OB B 



problem consists in determining the \'alue of Q ' ^ " " ' *■ in 



the equation 



fbi, V. . . . i|-, 1 ^ ;^ Tq ^1 e.,... 6^ ^ /OQ^ ae.. . . . ae\ X 



Accordingly I enunciate that 



Q 6', d.^... e, \ ^ ffi^^ a^,^ . . . «^,. I ^ fai^ a/^ . . . r//,. 1 



+ r^'/, «/,...«/. |x|«;t. %..-«/t/l . . (3.) 



subject to one sole exception in the case of ^,, 6^. . . 6^ being- 

 identical with 01, 02 • • • 4>r' namely, that for the terms (for such 



BR 

 case) of the form Q „^ a a) ^^^ value to be taken is not that 



t'l 6^2 • • • ^r 

 which the general formula would give, namely, 



but the half of this, i. e. simply the square of 



B B B 



The \alu(' of Q .' 2 • • • r^ j|. jg obvious, contains only (/uan- 

 9i 92 • • • 9r 

 titles of the form a^-b^, which are coefficients in the equations of 

 tranHformation, but none of the form n^.a^ or b^.b^; showing 



X 2 



