376 Mr Murphy on the Inverse Method of 



Thus if a represent the distance of any point, from the centre 

 of a spherical shell, the formula which represents the attraction 



both within and without (i. e. and — j is - -r- . 



18. To find a formula which shall represent /(a) when a</3, 

 and /(/3) when « is greater than (S ; /(<*) being supposed to consist 

 of positive powers of a. 



It appears from the paper above referred to, that the object 



1 <b (x) 



will be answered by taking the coefficient of- in -/'(.r)h.l. -*-— '- ; 



where f'(x) is the derived function oif{x) and (p(x) = (x-a).(x-fi). 



Thus, ify(a) = a m , vi being positive, we must take the coefficient 



of— in — m\\.\. (1 — ~ — ^r- ) ; that is, the term independent of .r in 



.1-'" V x(a + fl)] 



fl ( a(3 + xy ° _L ( af3 + x" \"< + 1 1 ( afj+xy ^* 1 



m \m{ a + /3 ) + x.(m + l)[a + p J + ^.(?» + 2) - U + /3 ) +&c -j> 

 which evidently is 



( («/3)" (m) ( a f*r + l (m + 3).(m) (a/3)"+ 8 1 



\( a + (iy + 1 •(a+/3)™ + 4+ 1.2 -(a + /3)'" +1 + KC }' 



the general term of which is 



m.{m + n + l).(?n + n + 2)...(m + 2n— 1) (a)3)" + ' 



1.2.3... w •(a + /3)'" + 2 "' 



Cor. Putting - for a, and -g for /3; the following formula will 



represent — when a>/3, and -^ when /3>«, viz. 



| 1_ w «/3 ?».(?« + 3) (a/3)' 1 



l(a + /3)- + l"(a+/3)" + * + 1.2 - (a + /3)'" + 4 KC 7- 



