360 Mr Murphy on the Inverse Method of 



If f(t), in the same interval, becomes infinite for a particular 



value of t (suppose when t=a), it is then of the form '-. — , P 



being a finite function of t, of which the greatest and least values 

 may be represented by p and p' ; <p (x) then is between 



*ljt=*r and ^'X(F^r- 



Now since 



r t' 1 f 1_ 1 1 m f 1_ _1 1 



Jt(t r a) m .v + l\(l-a) m (-a) m j + {x+l).(x+2)'\{l-a) m + l (-«)"+ •} +&c " 



it is obvious that when x = <x, <j>(x) = in this case also, with the 

 exceptions which we are now about to examine. 



When a = 0, then (J>(x) = f f - .f=f,P.t r - m , and putting for P its 



extreme values, <b(x) is evidently included between — — — = and 



x — vi + 1 



x -m + 1 ' anc ^ ' s therefore nothing, when x is infinite; but when 



m is not less than unity, it is clear that <p (0) = f— is infinite, the 



part of P not involving t, giving in this expression an infinite 

 integral ; this case is therefore inadmissible. (Art. 2.) 

 When « = 1, then <p(x) is included between 



if therefore m be not < 1, <£(0) is included between 



