240 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
/ 7* 
If — is negative, we find 
u 
= — - — 
, r \ 0 ' x 1 or J ' r \ 0 ' x ' x~ J 
X 2+V- — 
9 
oc % ~ ■ J ~ — 
9 
„ r 
If — = 0 , the solution is of the form 
9 5 
u 
- h { a °+ 5 +3 • • + { °s x ( b o + K x + h x2 ) } 
T 
If — is positive, 
u 
=?{ cos ( v/ k log*)(«o+5+3-)+ sin ( lo S*) ( 4 »+§+p-) } 
In all these the arbitrary constants are a 0 and b 0 , and the values of a l} b v a 2 , b 2 are 
determined by equations similar to those given in the former examples. 
Objections are commonly urged against the solutions of linear differential equa- 
tions in series, on the ground that the condition of convergency is fulfilled only 
within narrow limits of the independent variable. Might it not be shown that when 
a solution becomes divergent, there exists another which at the same limit becomes 
convergent, and that where no second form of solution exists none is needed ? 
In general the linear differential equation 
umu+fm^u . . . +f n (D)e*u= o 
has as many solutions in ascending series as there are simple factors in /^(D), and as 
many descending developments as there are factors of a like nature in/^D). 
B. § 3. On the Solution of Linear Partial Differential Equations by Series. 
Let x be one of the independent variables, u the dependent variable, and let the 
particular object proposed be to develope u in ascending powers of x. 
Put x=*J, and let the equation, supposed to be wanting of a second member, be 
placed under the form 
r Y^u-\- r Vffu-\-’Y 2 f 6 u . . .= 0 , . . . . . . . . (17.) 
wherein T 0 ,T 1 , T 2 are rational and entire functions of D and of the remaining vari- 
ables x',x", and of the symbols ^ 7 • 
Should it then happen that T 0 is of the form / 0 (D), not involving x', x" .. ^ 
we shall assume 
/ 0 (D)k=O, 
observing to introduce into the solution of this equation arbitrary functions of x', x", 
in the stead of arbitrary constants, and proceed with the result as in the cases already 
illustrated. 
