368 Mr Murphy on the Inverse Method of 



and putting »=1 to make both series comparable, term by term, 

 we get 



2B 4>C 62) „ A SB 5C „ 



x + 2 x + 3 x + 4,' x + 2 x + 3 x + i 



which gives B=A.\. C=B.%, Z>=C.f&c. 



2' 4 6 



so that <M*) = ^ + 1.^ + ^.^- 3 +& c.}; 



the constant .4 is found by giving # a particular value as 1, 



A f * 2 



which gives A = \. 



Ex. 7. Given 



, , . a.(a + «).(a4 2«) \a + (x— l).«j „ , .. 



* ( * ) = (, + fl).(, + /8 + a ).(« + fl + a«) {, + fl + («-l)«} to find ^' 



Following the same steps, we find here 



„.• B—JL[*-i). C=-f.(£- 2 )&c. 

 a \a I 2 In / 



whence f{t) = A.ta .(i_/)a . 



A similar method is to be used, when the relation between 

 <p{x) and 0(x + l) is of a more complicated nature, or when that 

 relation refers to more than two such functions. 



11. The denominators of the simple fractions in the two 

 preceding articles have been real ; such quadratic factors as would 



