EXPRESSED IN TERMS OF THE ROOTS OF (px, jx. 439 
X will be 2+ 14-^5 *• e. q~n-\-i-\- 2 , which is equal to q-\-i-\-e—m-\- 2 , the number of 
arbitrary constants in the most general values of AandL. Hence {A=H; L=X} is 
the general solution of A.y'4-L,(p=:A.r*+B.r*"'‘+---+K; and consequently the most 
general form of ... +K, which is evidently independent of the {t) arbi- 
trary quantities y*, will contain the same number of arbitrary constants as enter 
into the system (A) and (L), i. e. i+l. 
Art. (19.). Let us now begin with the case of greater simplicity when m~n, 
i. e. €=0 ; and let us revert to the system of equations marked (10.) in Section I., in 
which U and V are to be replaced by and (p. 
1st. Let 1, and therefore ^-f•L the number of arbitrary quantities in the 
conjunctive is w. 
From the system of equations (10.), we have for all values of ^2, g3...g„, 
(fiQo+f2Qi+ •••+^»-Q»-i)y’ 
— (flPo + ?2-Pi+ •••+^«-Pn-l)® 
= (§1 •Ki+f2*iKi+..-+f»-«-iKi)a?”-'-|-&c., 
and consequently the most general value of in the equation 
where ...-|-L 
will be obtained by making 
’'n-l 1 • 00 + ^2 -Qin- ••• 
ln-\— fl'Pfl f2*Pl”' fa-PreJ 
which solution contains w, i. e. the proper number of arbitrary contants. 
Again, if i=n —2 i-\-\=n — 1, which will therefore be the number of arbitrary 
constants in the most general value of ^„_2 of the equation 
‘^n^2v/” ln— 2 ’P~\~ S'„_2 — 0. 
This most general value of S^„_2 is therefore found by making 
’■n-2 = f\Qo + ?^2-Qi+ ••• n-Q/i 
Ifl-l— § iPfl f^2-Pl*** 
where f'l, f'2, are no longer entirely independent, but subject to the equation 
-K-i-f ^'2.1X1+ ... +f«-«-iKi=0, 
so as to leave (^—1) constants arbitrary. 
We thus obtain ^„_2=(f'iK2+f'2.iK.2+ ...+f In like manner, and 
for the same reasons, the most general values of in the equation 
In — 3 • 3 9 
will be found by making 
= • Qo +^2 . Ql + . • . . Q»- 1 
L-3— • Pq ^2.Qi . . . f». Pn-lJ 
3 M 
MDCCCLIIT. 
