REPORT ON CERTAIN BRANCHES OF ANALYSIS. 245 



none of which become zero or infinity, in as much as P does not 

 vanish when x — a. 



If there exist other terms in u of a similar kind, such as 



to' n^ 



Qix — by, R (a; — c)"", &c., the same observations will apply 

 to them. Such terms will correspond to values of x, which 

 make radical expressions of any kind zero or infinity, and the 

 form of the function u must be modified when necessary, so 

 that such radicals may present themselves in single terms of 



the form P (a; — a)». The same observations will apply to ne- 

 gative as well as positive values of — , unless we suppose — a 



negative whole number. The principle of the exception in this 



last case may be readily inferred from the remarks in the note, 



d~'' \ 

 p. 211, on the subject of the values of , _^ . — , when w is a 



whole number. If we suppose, therefore, « to involve terms 



m to' 



such as P (a; — «)», Q (« — 6)»', &c., the most general form 

 under which its developement can be put, supposing all terms 

 which become zero or infinity for specific values of x to be 

 rejected, will be as follows : 



, du , dUi h^ , d^u F 



TO TO 1 



a — a p, d« {x — «)" h 



(^-«)' ' dx-n ■r(i + ^' 



\ nj 



m' 



b — b „, dn' {x — 6)»»' A»' 



, du , d^u h^ d^u .h^ , o 



" ='^ + rf^^ + ^^iT2 + ^ 17273+ ^^^ 



+ &c. 

 We have introduced the discontinuous signs or factors — ;^— > 



