Definite Integrals, with Physical Applications. 359 



is made=0, be represented by A.^0), A>(0), &c. The equation 

 when both its members are expanded becomes 



and making x successively =0, 1, 2, 3, &c. we see that the cor- 

 responding terms in both series affected with their proper signs 

 are exactly equal, the two series are therefore identical for III 

 values of x. But the values of x from -1 to -00* are rejected 

 as, in general, giving infinite integrals, for then if A be the 

 absolute term in f{t) it is evident that U a f will be infinite In 

 the particular case where f(t) contains no positive power of * 

 below tr, we may then include the negative values of x, between 

 and -m. But in all cases when 0(0), tf>(i), 0(2) , &c . are 

 finite, we may assign any value to x between -1 and or be 

 tween and + ... and always have *>(*) -&$'.,. we sha „ 

 therefore in the following theory attribute to x all values included 

 between -1 and + 00*. and suppose *<*) to remain finite, during 

 that interval. 



3. Let us next consider the value of *(«) when x=x> } and 

 Jit) is any of the functions commonly received in analysis. 



If fU) always remains finite between the extreme values t=o 

 and t=l, let the greatest value of/(*) be represented by A and 

 the least by A', then <Kx)=f lA t).t< is included between f,A.r 

 and f,A',<, or between ^ and £-. it j s therefore =0 ^ 

 ■£ is infinite. 



