REPORT ON CERTAIN BRANCHES OF ANALYSIS. 245 



none of which become ssero or injinity, in as much as P does not 

 vanish when x ■= a. 



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



m' m" 



Q,{x — by, R (ar — cy^ , &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 V {x — «)». 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~^ 1 

 p. 211, on the subject of the values of -3 — zTr • — 7> when « is a 



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



such as P (a; — «)», Q (w — Z>)»', &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 : 



m m VI 



a — a p, d« {x — o)» h" 



(^^ rfj ■r(i + ^) 



m' m! m' 



h — h „, dn' {x — 6)"' Jin' 



d x^' 



(> - 5) 



or. 



, du J d^u h^ d^u .h^ ^ 



«=« + ^^' + ^^r:2 + rf^ 17273+ ^*^" 



m 



h — h ( Ifi \ Vi 



+ &c. 

 We have introduced the discontinnoiis sigtis or factors : 



a — a 



