THE THEOET OF EQUATIONS OF THE FIFTH OEDEE. 
273 
or it is 
=i2x{^a! d' —ly d ) ; 
and the third line is —x^ into 321^ — 2212, 
= 3(J'^-2«V)-2«V; 
or it is 
=x%Sa'd—ob''-). 
Hence, combining the three terms , 
(p + <p,<p, + (p^cpg = 2 Sa'd - Wd' + Id^- 
+x( 12 a'd'- 2 yd} 
+af( 8 a'd-W% 
or substituting for (a', b', d, d', d) their values, the right-hand side is 
=(40a^ 32«i, 28ac—8b^, 44«(?— 8^c, 28ae—A.bd-\-ld\x, l)h 
Avhere x stands for av, and on the left-hand side the factor cd is to be restored. 
25. Writing for shortness 
l)"=(40^r, 32«J, 28ac-8b\ ^^ad- 8 bc, 28ae-m-\-ld^\x, l)h 
the equation is 
«*((P,(P,4-(P.,<P, + <P3<P6) = (*^^3, ly; 
and the system of equations to which this belongs is 
a'(<?l‘p6+<p2<P4 + <p3<p5) = (*fe> 1)', 
^■(‘?^1<P2 + <P3^'4+<P5<P6) = (*X'^25 1)^5 
«'(<Pl<Po + <p2<p3 + <P4<P6) = (*fe. 
^'(<P1<P3 + (P2<P6+<P3<P5) = (*1^'I’4, 1)', 
®'( I ‘?4 + <P2<P5 + ^Ps^e) = ( *^^’ 5 : ^Y’ 
so that the roots .r,, x.,, X 3 , x^ will be rational functions of the combinations 
<Pi(Pg-l-<p 2 ^^ 4 -'i^ 3 ‘p 3 ? &c. respectively, of the equation in <p. 
2 p 
MDCCCLil. 
