380 Mr Murphy on the Inverse Method of 



and taking the coefficient of A" we have 



„ _ (2m + l).n.(n-l)...(ti-m + l) 

 (>i +l).(n + 2). ..{n + m + 1) 



and therefore 



M _ ft rr_ n ir _ ft rr (2m + l).m.(»l—l)...l . 



^„-0, /i,-0 ^„,_,-U, ^»- ( , w + 1) _ (;M + 2)- _ i(2w , + 1) ' 



_ tj (W+1)' rx _iy { (w + !).(« + 8) {' & 

 «. + ,-«.. j (2 , w + 2) . x:Im+2 - J "'"-i.2.(2 OT + 2)(2>» + 3) °"" 



The required formula therefore is 



1.2.3. ..m fl.3...(2?M-l) 2-'"(«/3)'" 



I' 



(»m + 1).(w« + 2)...2»«1 2.4...2»« '(en-/}) 2 '"-' 



1.3...(2w + l) (m + lf 2""" + *(a(i)' n + 1 

 + 2.4...(2»« + 2)'(2»« + 3)' (a + /3)"™ + 1 



1.3.5...(2m + 3) {(m + l).(m + 2)\- 2 2 "' + >(aP) m+ "- „ 1 

 + 2.4.6...(2ra + 4)'l.2.(2»« + 2).(2»M + 3)' (a + /3)-" , + 3 °'} " 



(2) Method of representing Discontinuous Functions of any 



number of Breaks. 



21. These particular instances have been put under forms best 

 adapted for many of their applications. We shall now proceed 

 to more general principles, for the representation of discontinuous 

 functions. 



To find a formula which shall represent 



<pi( a > & 7>"-)> «M a > ft 7>---)> <M«> ft 7»--) &c. 

 according as a, /3, 7 &c. are respectively least. 



