MinorDetermhiantsof Linearly EqimalentQiMdraticFunctioiis. 303 



not fail to strike the reader that it ought to admit of verification ; 

 for that U may be derived from Y in the same manner as V from 

 U if we express y,, y^- . • t/n iii tenns of Xy, x^ . . x^, by solving 

 the system of equations (2.), which there is no difficulty in doing. 

 In factj if we write 



yc^=acfi^x^ + a^^,w^-{- . . . +«2/3„.a;„ 

 we shall obtain 



^ o _ r«i «2 • • • «'-i ^''■+1 «'-+2 • • • ««"i ^ r«i '■'s • • • «« 



L^i, b^. . .bs-i bs+i b,+2 . ..b„J Vb^ h^.. . b„ 

 Accordingly we shall find 



fl»ii ttm.2 ■ . • dm 

 Opi (lp.2 • ' • Op, 



and 





Q "^1 "f 2 ■ • • "^Z^,- — / «'«1 «»% • • • «»«'• \ X ("^l ^ifo • • • «;>,• \ . 



o), eog • • • «,• ^/3^i ^^2 ■ ■ ■ l^ii>-' ^/Swj^wj. . . ^<^J 



substituting for the «'s and yS's their symbolical equivalents given 

 above^and applying the theorem given below, we shall easily obtam 



Qi/r, ■>lr^...^^__^(tmr+\ «JW,.+2 •••«»2„\ ^ f«-p,-+\ (^pr+2—»-pn\ 

 rUl &J2...&), ^b^,.+ -i b^,+^...b^n' ^fip,-+l^Pr+2-"^Pn' 



y~by b^. . . b„J 

 If, now, in the expression 



hi^bi^ ...bir-J L^d bg^...bQ/^b,p^b^.2'''Hr"<Pi^r"Hr 



we resubstitute for ^' ^2 " " ' ^'- w itg value in the form of 

 "<P\ ^'Pi • • • ^'t"- -^ 



« f ,~. Owj Ooij . . . Ou)r { 



we shall obtain ' '■ " ' ' ''' under the form of 



bi^ bi^ . . . bi^ 



