Homogeneous Quadratic Polynomials. 141 



this Magazine, we know that 



a J ~a^777ar-iar\ a■^ a^. • . «r-i • «r+i 1 



ff, «2 • • • ^r-\ «r «i flg • • • ^r-^ * ''^»'+' 



_ / flj «2 • • • «'-A X /'^l «2 • • • «r-l «r • «)-+l\ 

 \«1 fl2 • • • ^r-l/ V^l «2 • • • *'-l ^r ' ^r+i/ 



The first member of this equation is equivalent to 



/«! ffg • • • ^r-l «A X f"' ^2 • • • ^r-l ^^'M-A 



\a, flg • • • *>— I "r'' ^'^i ''a • • • *>— 1 ^'•+1'' 



__ /«, «2 • • • «'-l '^r Y. 

 \«, «2 • • • ''''•-l ^r+J 



Hence it follows, that if the two factors on the right-hand side 

 of the equation have the same sign, 



flj «2 • • • ^r-l «r J '''l ^2 • • • '"r-l «r+l 

 «j «2 • • • ''''—I ^J" «! ^2 • • • *"»— 1 ''''•+1 



have also the same sign nter se, and consequently the two triads 

 roi a^...ar-i~\ r«i ffg . . . «r-i «r~j r^i ac^'^ar-i Or ar+\'\ 



La, fl2"-«r-l-J Lffj Uc^. . .Ur-ia^J L«i ff2'"^'-l'^'-''^'-+l-' 



and 



ta, ff2---«'--l1 r«l «2'"«»-l «'-+l1 r^l «2 — «r-l «r+l «r"l 



will in all cases present the same number of changes and conti- 

 nuations, which proves that the contiguous umbrae, a^, a^+i, may 

 be interchanged without affecting the number of variations and 

 continuations in the entire series ; but, as is well known, any 

 one order of elements is always convertible into any other order 

 by means of successive interchanges of contiguous elements, 

 which demonstrates that, in whatever order the elements a, «2"' *« 

 be arranged, the number of continuations and variations in 



2 «i fliffa ^ ^ Wifls • • • «n 

 ' «l' Giff2 ' * * ai«2 • • • «„ 

 is invariable. But that the same thing is true (as we know it to 

 be), for the relation between any one of these unsymmetrical 

 series and the symmetrical series ;(vpsultipg from the method of 

 orthogonal transformation) ,,,^ \^,^^. ^,„,;+^V7«u \. . 



1, ^" ■, ■^ , . . . ^w , 



a, ajffa «i«2---«« 



is by no means so easily demonstrable in the general case by a 

 direct method, and the attention of algebraists is invited to sup- 

 ply such direct method of demonstration. My knowledge of the 



