BrcniiJcer — Errors Affeciing Logarithmic Compaiations. 467 

 Kow, since 



let the limit be put for C and the probable error for mF, and 



the value of W corresponding to the argument -^ may be 



sought in a table of this integral. 



If for example the greatest value found above (0.37) is put 

 for wF, and ? = 1, then the tables of the integral show for the 



argument ^-^ the corresponding value IF =^ 0.93, or, in other 



Vv^ords, under the most unfavorable condition (c = 90°) there 

 are 93 errors out of lOO which are less th^r* one second. 



B. As another example take the equation which is com- 

 mended in many mathematical works, 



sin'' 3^c = sin^ yz'a-\-b)— sin a sin b cos' y^C 

 Its solution is as follows ; first from ',he equation : 



cos i^C v'sin ci sin b 



C03/-1 = ^-=: . 



sin >2(«4-^) 

 the auxiliary angle ju is computed, then c from the equation 



sin }^c = sin }^{a-{-b) sin/i 



Since log cos m is computed by means of four logarithms taken 

 singly from the tables, we shall have 



/(log COS/.) = f\+Hf',+y2f'z-f\ 

 in which /'^ belongs to sin ^'2 (a~{-h). Thence it follows that 

 /(log sin M) = -cot^Mif\+yzr2+y2f\-f\)-cot'M/\+f', 



f (log Bin ^C) = -^/',-cotV(/\+3^/ .+f^/'3+/'l)+/5 



/(c) = ^ tan ^c . {/(log sin ^c)+/'s } 



