.358 Mr Murphy on the Inverse Method of 



SECTION I. 



PRINCIPLES RELATIVE TO CONTINUOUS FUNCTIONS. 



(1) Preliminary Observations. 



1. For the purpose of uniformity, and simplicity, the integral 

 limits which we take are and 1*. If t be any variable with 

 respect to which integrations are performed on any function, and 

 the limits of t are the finite quantities a and b ; then if we make 

 t=a + (b-a)t' the corresponding limits of the new variable, t' are 

 and 1. But if the limits of t are one finite and the other infinite, 



as a and ±oo, put then t=a± - — -, ; if they are both infinite as 



- oo and + oo make t= , /v_ 1 w generally, therefore, by these 



substitutions, and by more convenient ones in particular cases, we 

 may always make and 1 to be the integral limits. 



2. Let f(t) be any function of t, and make 



<t>(0)=f,f(t), <j>Kl)=f,f(t).t, <p(2)=f,f(t).t\ &C. 



none of which definite integrals is supposed infinite. Then, x 

 being any positive integer, we have ^>(x)=f,/(t).t T . Put 1-t' for 

 t (the limits of t' are also and 1, changing the sign of the 

 integral) and let the successive finite differences of (p(.x) when x 



* The limits uniformly adopted by Gauss. " Metkodus Nova Integr." 



