I 



142 Mr MURPHY'S SECOND MEMOIR ON THE 



which series when convergent will satisfy the equation jtf{t) . f' = ^ («) 

 for all values of x\ but even if not convergent, it will satisfy that 

 equation for all the integer values of x from to n — \ inclusive, 

 provided it be continued for at least n terms. 



If we multiply by f and integrate as before, we get 



which series when convergent may be used for any value of x, but 

 only positive and integer values when divergent. 



23. In Art. 21. when ftf(t).t'^(p(x) a given function of x, we have 

 found y(0 in a series expressed by functions of t of the same nature 

 as P„, now P„ is only a particular value of the general function (jo, q)„ 

 investigated in the former Section, Art. 12., namely, when p = q = l; we 

 shall now express /{t) according to this more general class of functions, 

 that is, under the form 



fit) = «o ip, q)o + «i (p, q)i + «2 {p, q)2 + &c. 



Now in Art. 12. above referred to, we have found 



, . _ {p + l){p + l+q)....{p^l + (m-l).q] m 

 Kp,qh-i l(l^q)....{i^{m-\).q} l"^ 



(2p + l)(2p + l+g)....{2jo + l + (?»-l).g} m.jm-l) 

 ■^ i.(\+q)....{\ + {m.-l).q} ' 1.2 '^ " *''• 



To simplify this expression, put 



77 = (/>^ + l)(M + l+9)--{p^ + l+(^-l)-g} 



'■'' l{l+q)....{l+{m-l).q} 



Let yj^ express the operation of changing h into h + 1 (Vid. former 

 Memoir, Note B. 2.), >//^ the repetition of this operation a second 

 time, &c. ; the preceding series will then become 



