XII 



THE THEORY OF THE LONG INEQUALITY 



, 2 A> (I) 

 and a -*T 



&c, by the formulaB, 



s d%(i) ,d%JV 2k + 1 ( t d>b k ® t *b k+l <i> 



da 2 



. d'h^v gfc + l / t 

 ° da 2 (1 - a? V da' da* 



4a ( db k U) _ d6 fc+ /i)\ 



) 



4a / i 



a- 



1 -a V 



+ a 



da 



2a 2 



(!-«)« 



;(**<» + **+,<»)> 



.d"6fc (l) ,«T6 fc+1 «) 2*+l / 9 d*6 fc «) cP6 fc+1 «>\ 



da- 



da 2 



4a f db k M db k+1 dh 



4a / 



a 



1 + a\ 



da 



— a- 



da 



2a s 



(1+a) 



all which are deduced by differentiating the formula? for b k ^> and 6 fe+1 ^), 



dbP t dVi) . ^.a) . ,d 4 6 B (i) 



The values of 6,(D, a ~ 



da da 



of the series 



6.® = 



3.5.7. 9.11 



da 3 

 13 



were also verified by means 



'-±- and a 4 



da 4 

 1.3 13.15 



4.6.8.10.12 a ^ 1+ I-Ii« a+ 2TI-I4Tl6 a4 + &C ^' 

 and its differential coefficients. 



12. The values of the fundamental quantities are 



a = 0-638647761 = sin (39 0, 69106023) 

 log a = 1-805261394, 

 6 ® = 2-27020865, 

 6 2 © = 0-37818958, 

 6 4 ® = 0-11459739, 

 6 S ® = 0-03888697. 

 6 (l) = 6-3025242, 

 6 2 (*) = 3-8841368, 

 fe 4 (i) = 1-9712274, 



a __°_ = 0-73084947, 

 da 



a—^— = 0-96086031, 

 da 



d6 4 © 

 a—— = 0-52631179, 

 (ta 



d6 6 ® 

 a - !— = 0-25732242. 

 da 



da 



