t 478 ] 
PROBLEM IV. 
Suppofe ABC /(9 be a fpherical triangle^ in 
which are given the two fides ab, bc, 
with the included angle b, to find the 
-third fide ac. 
A 
B 
SOLUTION I. 
Let ABC = ft bc = AB = (5'. Put AC = /3 + f, /S bciiig 
an approximate value of ac, when the two legs are 
nearly quadrants. Now the cohne of ac being equal to 
cof. ^ X cof. cx. X tang. y/S-cot. ^Sx vf. f. Therefore f is the 
difference of two arcs whofe fines are cot. /3x vf. and 
cof. (5' X cof. a X tang. Y ft the difference of thefe two arcs 
being diminifhed by the corredtion cot. ^ x vf. f. 
('a j It is a well known theor. that fin. ba x fin. bc ; zz vf. ac — vf. ab — bc : vf.B ; 
that is, fin. ba x fin. bc : — cof. ab — bc — cof. ac : r — cof. B. Or, 
in the author’s notation, putting rzzi, da: i rrcof. ^ — » — cof. ac : i — b. 
Therefore da — ^DA=cof. ^ — «— cof. ac. Or, cof. Acrr^DA — da— cof .^" — ». 
For cof. ^—a. fubftitute its value as exprelTed in the fecond corollary of the 
lemma, and there arifes the author’s equation, cof. A.c—bDA-\-da, 
b'DA+da^“^ we fhall have b~DA + da: and 
S, HORSLEY. 
EXAMPLE. 
