Dremiher — Errors Affecting Logarithmic Computations. 40.) 

 Tlie order in v/Iiich tho loi;arithni3 are writ'ten is as follows: 



1. sin y'ya—b) cos 3'4C 



2. sin X-^ia^-b) oin y^G 

 cosu 



o 



4. 



5. cos 3^(a — h) cos 3^(7 



6. cos >^(a-f-6) sin 3^C7 

 sinr 



sin l^G 



7. 

 8. 



cos YzG 



Lastly fi-om the diiierenee of the logarithms of 4. and 8. arises 

 log tan yo ^j ^y means of which the side c is found. But since 

 the anples u and '^ , which are used h'*re only as auxiliaries, 

 have lilt a small influence upon the probable error by which c is 

 aifectfd, as already has appeared in example A, the formula 

 may }-r- sim.plified. As the transition has t-aken place in angle 

 H froji'i tangent to cosine, in angle v to sine, w^e shall w^rite in 

 place of them 0° and 90° respectively. ISTevertheless we could 

 pursue a different method vvithout causing any essential differ- 

 ence in the final formula. 



ij'urthcrn:ore, if the errors by which log cos 1/^ C and log sin 

 1/2 C are affected be denoted by f\ and /' 2? and the errors of the 

 other logaritlmis, which are used only once in the computation, 

 by /'s, f\, etc., we shall have 



/(log tan fi) =f\-f'',±f\—f\ 



/(log cos /0 = -3in^"(/l-/'2+/'3-/'4)-sin2/^/^+/'5 



/(log sin 3^0 =/'2+/'4"/(log cos /x) 



then 



/(lo^ tan K) =/'i— /', l-Z'e— /"t 



f{\o4, yin v) = Q)rr{f\-f2^f\—f' ^)^co3'y f i—f\ 



/(log cosj'o^j) =f i-h/'e— /Cog sin v) 



lastly 



/{log tan>^c) = -/'i+/'2+/'4-/'6-/(log cos//)+/(log sin r) 



■whence it follows that 



/•(c) = — sine {/(log tan ^0)4-/% ( 



