relating to the Products of Sums of Squares. 517 



w,i, a,,, b„, Cji . . . being determinate functions of w, a, b, c, . . . ; 

 w„ a,, b,, Cf . . . homogeneous of the first order in the quantities 

 of each set ; the value of Wn being obviously in every case 



Wii=siiviVi—aai — bbi — cCi . . . 



and a,i, b,i, c,i . . . containing in every case the terms aWi + a,w, 

 bw, + biW, cWi + CiW, . . . but the fonn of the remaining terms 

 depending as well on the triplets entering into the system of 

 quasi-equations as on the values given to the signs + ; the 

 quasi-equaiions servmg, in fact, to prescribe a rule for the forma- 

 tion of certain functions iv^, a,^, b^, c^j. . . , the properties of which 

 functions may afterwards be investigated. 



Suppose, now, that the system of quasi-equations is such that 



being any two of its triplets, with a common symbol e^, there 

 occur also in the system the triplets 



/o«o^o^ fo^o^o'' 9o%^oy goK^^oy 

 and suppose that the corresponding portion of the system is 

 eo«o*o = e, eoCado = e' 



goOodo = h ffoboCo = ''', 



where e, ^, i, e', ^, i! each of them denote one of the signs + or 

 — ; then eiijfii, ff,, will contain respectively the terms 

 €{abi — afi) + e' {cd, — c^d) 

 {{ac,-a,c) + ^'{db-dp) 

 t[adf—a,d) + J{bc—bf) ; 

 and «/+//+y;/^ contains the terms 

 {a'' + b'' + c' + d^)[a^ + b^ + cf + d;\)-a%''-b%i^-c'c^-d^d^ 

 + 2 [ee' [ab, — a/b) {cd, — c^d) 

 + ^^'{ac,-a,c){db,-d,h) 

 + u' {adj — a,d) {be, — bfi)] . 

 And by taking account of the terms ew, + ep, fio, +f,w, gw,-{-g,w 

 ine„,f„, //„ respectively, we should have had besides in e,^ -{-f^ +g^f 

 the terms 



{"^^f^+f-^f^^f-^n^o^"- 



+ 2{ee,+ffi-\-gg;)ivio,. 

 Also w/ contains the terms 



iv'^w,^ + aX + l^V + ^V + d^d^- 

 -2{cp,+ff, + gg;)ivw,; 



