296 Mr. J. J. Sylvester an the Relatim Metweeu ihey\ , \v\>,\«. 



syzygy in question, which, as I indicated, is Unear ; by which I 

 mean that a determinant of the one function is equal to the sum 

 of the pari-ordinal determinants of the other affected respectively 

 with multipliers formed exclusively out of the coefficients of the 

 equations of transformation. In order that a clear enunciation 

 of the theorem in view may be possible, it is necessary to premise 

 a new but simple, and, as experience has proved to me, a most 

 powerful, because natural, method of notation applicable to all 

 questions concerning determinants. . , 



Every determinant is obtained hy operating upon a square 

 array of quantities, which, according to .thcr ordinary i method, 

 might be denoted as lolloWs : _ 



"n, 1 "«,2-* • • "»,« - ' ^ 



INiTy method consists in expressing the same quantities Dilated 



rally as below : fifnuijn'iot-jb hnuoqmo^ oai fsransg xii bnA 



./ - '■'2*1 ^2*2 • • • *»i^« ,/ 





1 «„«2 • • • "n.'*,0 



onsb ll'm 



where of course, whenever desirable, instead of a^, a^\ . hn, ato2 

 «i, Wg . . «„, we may write simply a, b . . I, and «, p .''\' a, re- 

 spectively. Each quantity is now represented by two letters j, 

 the letters themselves, taken separately, being symbols neither 

 of quantity nor of operation, but mere umbr?e or ideal elements 

 of quantitative sjanbols. We have now a peaps of representing 

 the determinant above given in a compact form; for this purpose, 

 we need but to \yrite oi^e, set pjf UTOibyje qV|e^' tjie pthej* as fpUp^iy^,;, 



/«! </2 ... fir„\ jf^re'ri'owAvish to obtain the algebraiic Viiti(i' 

 \a, aa . . . aj " ,; / . 

 of this determinant, it is only necessary to. take qtj, u^ixjii.ekd:.i&' 

 all its 1 . 2 . 3 . . »i difl'erent positions, and we shall have 



fa, (Icy . . . a,,\ ^ , , , , ., ■ 



< - " >=X±{o,ug +a,ug + . . . +anUQ^}, 



Laj «.j . . . «„J , ' , 



' . ■ ■ ■ \. -_. 



in which expression dy 6^ k > • 0^ represents some order of jthe 

 numbers 1, 2 . . . ti, and the positive or negative sign is to be 

 taken according to the well-kno\vn dichotomous law. Thus, for 



example "' '"' •J"-"''' iii'mjuJ jnhjiUUilu -rmi ,tv.." iiihiU i ts i*>.i lllii 



. ' [(-.«[ \vs(\k .|. .oY , [ ,)oY J y. ^^ft\A \n\^ 



