760 PEOFESSOE BOOLE ON THE COMPAEISON OF TEANSCENHENTS, ^TH 
Or if we appropriate the symbol h to express complete ditferentiation with respect to 
^25 • • 
"N^’hence 
A^^herefore 
^dx+lE=0. 
7 SE 
*=-jE- 
{90 
dx 
'2Fdx= — 2 
F§E 
'dE' 
dx 
( 10 .) 
FSE 
Now F being a rational, and E a rational and entire function of x, the expression 
dx 
Avill be rational with respect to x, whence, by the subsidiary proposition just demonstrated. 
Therefore 
^ d^ — — © 
dx 
^gE~ i^logE 
dE 
dx 
^ (m 
dx 
dx 
dE 
dx 
E" 
( 11 .) 
Now the distinctive part of the performance of the operation 0 in the second member, 
consists in developing the entire function 
F— V ^ or F — 
^ rfE ^ E ’ ^ E 
( 12 .) 
dx 
in ascending powers of certain simple factors of the form x — those simple factors 
being, 1 st, such as are found in the denominator of F ; 2 ndly, such as are not found in 
SE 
the denominator of F, but are found in the denominator of 3 =. It mav be shown that 
dhj 
the result of that portion of the operation 0 which depends upon the latter class of 
factors is 0. For the only factors of the form x—p which produce terms of the form 
in the ascending development of ( 12 .), and which are not found in the denominator 
of F, must be found in E. Let x—p be any such factor, then w^e may write 
E=H(^— j))™, 
where H does not contain x—p. 
Therefore 
8E (a?— — (a;— jo)8H — jhHSjo 
dx 
(13.) 
