768 PEOrESSOE boole ox the compaeisox op teaxscexdexts. ^th 
The partial fraction which has v for its demominator will be Separating this, 
the entire fraction (6.) -wdll assume the form 
Thus (5.) becomes 
q v'q /“ m ® — 
2 « '!-dx=.nQf- 
q V q V ' q t u — s 
(-•) 
0 deriving its interpretation from q and n. 
The first term in the second member, inasmuch as we may give to it the foiun 
phu 
nO 
qv 
vanishes by virtue of (6.), art. 11. The second term may be reduced to the foim 
p s”‘v'^~^du — s'^uv^~^tv 
nQ 
■ - 
J 
( 8 .) 
ftnyti — gm^n 
In proof of this, I observe that the function to which 0 is applied cannot have any of the 
simple factors of v in its denominator. No such factor is involved in q. For by suppo- 
sition all the coefficients in q are definite, while those in are arbitrary. Neither, again, 
can any of those simple factors be involved in s™!;”, for if so it will be involved in 
and therefore either in t or u. But it is not involved in t, as the constants in t are 
definite ; and it is not involved in for if it were - would not be in its lowest temis. 
V 
The expression (8.) may be written in the form 
^ s\”‘ vdu — udv 
nQ\ 
VJ 
8 
1 
or 
23 o U 
on replacing -? - and -hj(p,-ip and 
Hence 
Therefore 
(9.) 
The symbols 0 and J in the second member are now independent and may be trans- 
posed. We thus have 
( 10 .) 
an expression of remarkable simplicity. 
21. In applying this theorem Ave must effect the integration in the second member, 
regarding x, a,s the only variable, inasmuch as the variables Uo, b^, See. enter into the con- 
stitution of but not into that of the other rational fractions <p and %//. When the 
