164 MR WALLACE'S Formula for finding the Logarithms 



By comparing the latter two expressions with the former, and 

 observing that A being an angle, sin 2 A 2 sin A cos A, we get 

 these formulae, 



cos i x sin (x + A x) , 



A log sin A x = TT - -, - . - r^ . , ' - r A log sin x ; (G) 



2 cos (x + | A x) sin \ (x + A x) 



cos x cos (x + A x) , 



A log sin \ x = -. - ; - r -. : - : f-f - - ( A log cos x) ; (H) 

 4 cos (x + A x) sin ^ x sin ^ (* + A ;r) v 



sin i # sin (# + A a?) 

 -Alogcosi-r == g^i Acos x/ Ag * log sing; (K) 



cos a; cos (x + A a?) 



A loT COS i a? == -: - 7 - ; r~, : - r -- ; - TT - ; - r A log COS X . (L.) 



4 cos (x + ^ A a-) cos a? cos (x + A a-) 



16. These expressions exhibit elegant analytic formulae, but 

 they cannot so readily be applied to calculation as the other for- 

 mulas, on account of the factor cos (ac + 1- A x), which enters 

 into them all. If we assume that 



cos (x + I A x) = V cos x cos (x + A x) , 



which is nearly true when A an is small, they become more simple ; 

 however, they are less accurate. When put under the same 

 form as the expressions for A log sin so, and A log cos x, they will 

 stand thus, 



( cot I x sin (x + A x) A I g s i n x \ sin I a; 



' - "V - 



(coti x Vcosd? cos (x + A*') A log cos x) sini;r 



log sin 4 = < - - . - - --- 5 - f . , , - r-; (ti) 

 ( sin a? x, ) sin (x + A x) 



( tan x sin (x + A x) A log sin x ) cos 1 x 



- - 



CO. 





f Vcos # cos ( ^ + A #) A log cos x) cos i a? 



A log cos 1 x = < ~ - i - 1 . - &-. - >- - ^-S - . (L ') 



( cos 2 \x 4 j cos i (* + A x) 



Of these, formula? (H 7 ) and (L'), which give the increments of 

 log sin x, and log cos ^ x, from that of log cos x are quite ana- 

 logous to those investigated in the beginning of this paper, and 



