280 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
2nd. If the coefficients <p 0 (tt), ^(t), &c. are constant quantities. This happens in 
the well-known class of equations 
u x -\- a M^')u x - l +o, 2 (p(jc)(p(d'- l)te,_ a +&c.=U, 
which becomes 
(1 -j-« 1 ^+a 2 ^ 2 -}-&c.)w ;r =U J 
and is integrable by resolution of the operating factor, as in equations with constant 
coefficients. It may be worth while to note that this class of equations, when of the 
nth degree, involves one arbitrary function <p(* v), and n arbitrary constants, a v .a n ; 
but the preceding class under the same circumstances involves one arbitrary function 
with n-f-1 arbitrary constants, viz. h with n constants in f 0 (x). It is therefore the 
more general of the two. 
3rd. If the equation should consist of only two terms. Suppose that it should be 
reducible to the form 
<Pa(*)u+<p l (*)g n u=U, 
which may be put under the form 
u-\-p(‘7r)^ n u=\J 1 (105.) 
Proceeding as in the corresponding case of C. § 2, it may in all cases be determined 
whether the equation is integrable or not. 
In general the equation u-\-cp(‘r)g n u=XJ can be reduced to the form 
by the relations 
_T> PM 
rf* 
u=p 4 t!v, 
(106.) 
precisely as in differential equations. 
Thus, to pursue the analogy, the equation 
q n 
W+ , — : w — 1 — i CP W W = U, 
1 (7r + <7 1 )(7T + ff 2 ) . . . (7r + G n ) 5 
in which a Y a 2 . . a n are in the order of magnitude, can be reduced to the form 
and then integrated by resolution into a system of equations of the first order, pro- 
vided that the quantities 
a l — dc l —\ a l —a 3 —2 a l — a 4 —3 a 1 — a n —n + l 
n 5 n ’ n ’ n 
are all integers. 
Ex. Given {x 2 -\-ax-\- b)u x -\-(2x—a — \)h(p(x)u x -i+(h 2 —q 2 )<p(x)<p(x— l)w*_2=0. 
d_ d_ 
Let v=x— n<p(x)e~dx } g=<p(x)e~dx, then 
(x 2 -\-ax-{-b)u-{-(2x—a — 1 )hgu-\- (Ji 2 — q 2 )g 2 u=- 0. 
Now x—%-\ therefore 
x 2 -\-cix-\-b = 7T 2 -f- an -f b-\- (2-r — a — 1 )w£+ra 2 f 2 , 
2 x — a— 1 —'Ik— a — 1 -f-2 ng, 
