/: 



Applications o/' Abel's Theorem. 123 



If this series be called /7, the integral 



I find from this series for the values of the integral 



from :f = to ;r = 2, \I/' '49695 







X = to a; = —2'SSTn, -^ —■ + -34.586 



X = to a: = 5, ^/'-^ — -199968 



X = to or = —5-3864.5, —4'' -^ + '185628 



In order to have a convergent series for the value of the 

 integral from ^ = to ^ = -88721, I make 



4 



X 



/ 



dx _ ^, i , 3z/| 3.72/| . 3.7.11?/ V' 



\/l+x'' 4.5 ^4.8.9 ^4.8.12.13 ^ 



and I find for the integral in question, the number -f- -84288. 



The integral from ^ = to ^ = -38645 may be found at 

 once from the expression 



dx x^ 1.3 a:'' 1 . 3 . 5 ^'^ 



v/l+^* 2.5 2.4.9 2.4.6.13^ 



/- 



and I obtain for this integral the number + -38561. 



It appears in this case that the constant on the right hand 

 side of the equation equals — 2 \(/'i, and 



•49695 + -34586 — -84288 = nearly* 



-19968 + -185624 — -38561 = 



according to the general theorem. It may be proper to men- 

 tion, that the transformations adopted in order to procure con- 

 vergent series for the integrals required, are all taken from 

 Legendre's work. 



The equations of condition between the quantities x^, x^, x^ 



&c., may be varied to an almost indefinite extent: those 



which I have adopted in the last example are the same which 

 were used by Mr. Talbot, and through which he obtained a 



* The theorem is of course rigorous, but it can only be verified ap- 

 proximately in numerical examples. 



R2 



