304 COMPUTATION OF THE VALUES OF LOG '," 



5. Taking p. = cos and p.' = cos &, the following formula is convenient 

 as a test of accuracy in the determination of /x p.'. 



p. - p.' = cos {1 - cos (0' - 0)} + sin sin (0' - 0) 



sin sin (0' 0) + cos ,-& / 



1 + cos (00) 



sin 2 0' cos 3 



= e 2 sin 0' sin cos + 



.././] /i, 



= e- sin 0' sin cos 1 + - 



l+cos(0'-0) 



e 2 sin 0' cos 2 







- -Q-T r^ ^ . 

 sm0{l+cos(0'-0)}J 



Also sin $ = e sin 0' cos 0, 



hence u.-u! = sin i// sin 1+ e 2 cos 2 ( -- - ) 



r \l+cost///J 



6. The following theorem is also useful for testing the accuracy of 

 numerical determinations of similar functions of p. and //. 



Let G be any function of //., and G r the same function of p.'. 



Since /A = /A' + (fji p.') and // = /t (/x /A'), we get from Taylor's Theorem, 



dG' 



1 , . id'G d s G'\ 1 t fd'G d 4 G' 



- + d?) ~ 24 ~ 



G dG 



But by the same formula 



d?G d*G'\ . .kl*G d*G 1 



d*G 



, ,, , 



and 2 - - -5-7^ =(^-JM') -- + T-^ +&c. 



4 ' 4 n x 5 ' 



