170 MR WALLACE'S Improved Solution of a Case, 8fc. 



To find side c by the common method. 



sin A 9.9870362 sin B 9-8354297 



To find the side c by the method here proposed. 



cos i (A B) 9-9818861 sin (A B) 94516338 



cos i (A + B) 9-7037054 sin $ (A + B) 9'9359289 



a + b 5-4611938 a b 4-6987180 



15-1648992 14-6346469 



c 152409-8 5-1830131 c 5-1830131 



5. By comparing the two solutions and their verifications, it 

 appears that by the common method, the Trigonometrical Tables 

 must be opened in ten different places, and that by the new me- 

 thod, only five different openings are necessary, because in this 

 last, the logarithms of a + b and a b occur twice, and the 

 cosines and sines of the angles ^ ( A -f B) and ^ ( A B) are 

 found in the same lines with their tangents. Even if we are sa- 

 tisfied with a single solution, omitting the verification, the me- 

 thod I have here proposed still appears td deserve the preference, 

 because, in the common way, the tables must be opened in eight 

 places, but in the other way, only in five. Of course I do not take 

 into account the operations necessary to find the sine and cosine 

 from the tangent. If they be found by formula (F) art. 12., of 

 the preceding paper, only one opening more will be required, viz. 

 that for the logarithm of the difference of the log. tangent ; and 

 it is presumed that they may be found more readily than the 

 correct values of ^ (A B), (which, in general, is only a sub- 

 sidiary angle, and not wanted in the solution), and the log sines 

 of the angles A and B. 



