Definite Integrals, with Physical Applications. 361 



and is infinite, the absolute term in P giving as above, an infinite 

 integral; rejecting therefore, this case, and supposing m<\, if we 

 integrate by parts, we find 



r t' _ X r t" 1 



Jt(t- 1)'" ~ x + (l-m)' J l (t-l) m; 



this integral therefore converges to 0, as x increases to infinity. 



Suppose next that f{t) becomes infinite for several values of t, 

 as t = a, t = a, t = a", &c, 



then/(0 is of the form (t _ a) ^ {t _ d ^ {t _^ r _ -> 



which being resolved into simple fractions, the same reasoning as 

 above shews that for all the admissible cases, we have <j>(x) = o 

 when x = oo . 



Lastly, let f(t) be imaginary, so that f(t) = P + Q *J — 1, P and Q 

 being real functions of.£, then (f>(x) = f t P.f , + x /-lf t Qt r , and by 

 the above reasoning, both the latter functions vanish, and conse- 

 quently <j)(x) also =0, when x=cc. 



4. From this examination we see that the inverse problem 

 "to recur from 4>(x) to f(t)" consists of two parts*, first, when 

 <p(x) converges to as x increases to x, and is therefore essentially 

 composed of negative powers of x; in this case which forms the 

 subject of Part I. f(t) is of the form of the usual functions re- 

 ceived in analysis; secondly, when $ (x) does not consist of such 

 powers, for instance when it is always zero, and then f(t) belongs 

 to a new and remarkable class of functions which will be treated 



•This is the division of Integrals into large classes, alluded to in p. l.'i<), Vol. 111. 



Cumlj. Trans. 



