MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
249 
y(D) 
If <p(D) involves any factor of the form 
let ?(D)= X p+^.y x'fD), and let ^(D)=^'(D), then 
it may be made to disappear ; for 
p 4>( p )_ P %( p ) 
%(D)~ *\(D ±ir)’ 
which is finite. 
From inspection of the above it is evident that if <p(D) is in the form of a rational 
fraction, and it is proposed to diminish (so to speak) D in any factor of the numerator, 
or to augment D in a factor of the denominator, by a multiple of r, the process bv 
which u will be finally deduced from v will depend upon differentiation ; but if it is 
proposed to augment D in a factor of the numerator, or to diminish D in a factor of 
the denominator, the process will involve integration. The former is obviously the 
preferable condition. 
The general proposition (XIV.) amounts in reality to this, that the equation 
u-j-<p(D)z r<, u = XJ 
(29.) 
may be resolved into the system of equations, 
.. _ ~n 0(D) 
u ~™ r ^D) v ’ 
U — p ^ p )y 
u—r ^(D) v ’ 
v+^(D)z r y~V, 
(30.) 
(31.) 
whence V, v, u are to be successively determined. Of these equations we shall call 
the two first the auxiliary ones, and (31.) the transformed one. This premised, the 
following are the canons which regulate the determination of the constants. 
1. If no factor of <p(D) disappears in ^(D), no arbitrary constants are to be intro- 
duced into the solutions of the auxiliary equations ; those derived from the trans- 
formed equation being necessary and sufficient. 
9 
Disappearing factors are in general of the form 
D + a 
D + 6’ 
a — b being a multiple of r. 
Every such factor will give a system of ° L ~ constants in the solution of one of the 
auxiliary equations ; if in that of the equation determining V, those constants will 
be arbitrary, but one only will need to be retained ; if however in that of the equa- 
tion determining u, one only will be arbitrary, and the rest will be therewith con- 
nected by the relation u m -\-<p{m)u m _ r = 0, derived from the primitive equation. 
The reason why the constants connected with the disappearing factor are arbi- 
trary in V alone, is, that V enters into no other equation than the one in whose solu- 
tion those constants are found. If however the entire series of constants in V are 
retained, they will be reduced to one by the subsequent differentiations in passing to 
the value of u. 
Ex. 1. To determine the general characteristic of those differential equations of 
the rath degree, the solution of which depends on that of the equation 
d n v 
chi’ 1 
±q*V = X. 
2 K 2 
