THELR APPLICATION TO THE SOLUTION OF EQUATIONS. 217 
obtained in the second section may be considered as given 
d priori by Lagrange’s theory. 
Lagrange arrives at a reducing equation which may be 
written 
S$-TS+4+US -X34+Y=0, 
where $=$§/(2”")??, and the coefficients T, U, &c. are, each 
of them, susceptible of only six different values, and Y is 
obviously the fifth power of the expression denoted in the 
above Paper by 6. Now if, in seeking these coefficients, 
we commence with T, we shall be conducted to functions 
of five dimensions with respect to z. Thus, developing 
S={fOPH=(Mti mtr Pate ate as) 
according to powers of 7, we have 
S=8 461+ &0 +E P+ &%, 
where, employing (for convenience) the new cyclical 
symbol, 
£=Sa° + 10S! {204 (wy +24 2,) + Bxy(a}a?-+23.22)} 
+1202; 2423 2,2, 
£,=5> (a4 x, + 2a} at + 4a} wy v7, + Gaj xy a5+ 12a} x, x52), 
E,= 5D (aj wy + 2a} ay t Aaj vy 25+ Gary, &, a+ 12x L2 %5 45), 
E,= 5D! (vf vy + 2ay v3 + Ar} ay Hy + Ox} Hj Xs + 12a} x, Hy 25), 
E52 (vt x; + Qa} a3t 4a} vy X14 Gay a3 x34 12axj 7, x, x5); 
and if we distinguish this value of 3 by 3(7), the others 
will be denoted by $(#”), S(#) and 3(2*). Consequently 
TH=SS=46,4+(&+&+84+8&)G+P +8 +H) =58, — Zé; 
or, since 
DE=Lx’ +5ret 2.4 lOXa} x} + Wea} x, 7, + 80Le7j a} x, 
+ GOD 27 ay #3 +1202, 2 ¥3 2,25, 
T= 42a — 5d 2% x, — 10223 23 — 202.2% x, x3 — 30227} 43 25 
— GOD a? ay 5%, + 50S’ § Qa} (ay v; + a3 %4) + 8a, (a3 v2 
+ £307) t +4800, Ly %3 Vy 5. 
Applying the =’ process (Sec. II., Art. 19), 
LS! a}( &ya5 + & gy) = Ef} (dats + yy) (LH — Ly— Ly— Xy— Xs) 
= La’ ai (ayes + aya) — Lay (yxy + VyHs + hag t+ U5; 
H Wy Ughy + UyV Ly + LyX Xs + Vy) 
