continued Fraction for Circle-quadrature. 333 



and putting m=l, 



81og2=l + | 1.8 (a) 



and so on. 



But, secondly, by expanding y m ~ l {l + y)~ l in a series of ascend- 

 ing powers of y, and integrating with respect to y between the 

 limits and 1, we obtain 



2m 



f>-'(l+y)-^=2(l " + *_*+.. A 



L y v y ' y V m + 1 m + 2 m + 3 / 



Now the series here enclosed in brackets may be looked on as a 

 particular case (viz. a = m, /3=1, 7 = w + l, oc=— 1) of Gauss's 

 general series F(a, /3, y, <r), which, owing to a special property, 

 he was able to expand in the form of a continued fraction. 

 Taking advantage of this, we find, on making the necessary sub- 

 stitutions, 



2m\ y m - l (\+y)- l dy 

 Jo 



and, as before, putting m = ^, we derive the well-known result, 



1 + 3 + f. 32 



5+ 7 + .._ 



and putting m = l, there results a companion identity to («), 

 and so on. 



Thirdly, the series found above may be expanded in the form 

 of a continued fraction by means of the ordinary general method. 

 Doing this, it is found that 



f 1 2 



2m y-'ll+yl-^j « ; )2 (HI.) 



+ 1 + . 



