244 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
u — 'Zu m z m6 — '%u m s m , 
m(m-l)u m — (fff-fft))(m-\)u m -i- (^+(/i(^+/ 2 ( 0 ^+/ 3 ( 0 ) m »»- 2 =O. . ( 20 .) 
The equation m(m— 1)=0 gives m = 0 or 1, whence u 0 and u x are arbitrary functions 
of t, which we shall represent by F 0 (£), Fj(£). The value of u m , which we shall simi- 
larly represent by F Jf), is given by (20.), whence the complete integral will be 
m =Fo(*) — Fi(05+F 2 (^ 2 4-F 3 (0a 3 • • • ad inf., 
wherein s=x—y, t=x-\-y, F 0 (£), F 1 (#) are arbitrary, and in general 
w -/ a (0)F.-.w + (£+ cam +/ 2 w)^+/ a w) 
F ' m (0 m{m— 1) 
The derivation of the coefficients is thus always possible. 
B. § 4. On the Integration of Linear Differential Equations in Finite Terms. 
If we affect both sides of the equation 
/o(D)u+/i(D)A* . . . +f n (T>)z nl >u= U 
with {/q(D)}- 1 , and for write <p x , (D), p 2 (D).., and for {/ 0 (D)}- 1 U 
write U, we have 
m+^ 1 (D)s^m . . . +p„(D)s n *M=U ; (21.) 
under which form the linear differential equation will be treated in the following 
investigation. 
We however premise the integrability of equations of the form 
F (/«s)“= u - 
ct d /* dx 
for, writing > whence t—J we have 
f G 4)“= u > 
which, for the forms of F here contemplated, is an equation with constant coefficients. 
The linear equation of the first order is an example of the above class, for, writing 
it in the form 
d d 
we have only to assume in order to obtain 
f'dx du _ 
.-. u =i --fiQdt=rff^%d x . 
