PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 129 



28. We may employ two subsidiary angles A and E, one of which 

 may be taken at pleasure. 



Fifth. Supposing E to be any known angle, let A be such that 



(5). 



sm \1/ . 

 sm A = -. — ^ sm E ; 

 sm 



then ^'^"i (</'-'/-) _ tani_(E-A) 

 ' tan ^{(p + ^) tan 1(E + a)_ 



Sixth. Taking E as before any angle, take A such that 



sm ^ 

 cos A = ~. — ^ cos E ; 

 sm (p ' 



, , tan i ((6 — \^\ 



^^""' tSr|l^) = t^»i(E + A)tanX(E-A 



(6). 



Seventh. Assuming E any angle, find A so that 



, . sin \1/^ , 

 tan A = -. — i- tan E ; 

 sin ' 



then, tani((^-v/>) ^ sin (E-A) 



tani(0 + V/) sin(E + A) 



(7). 



29. These formula determine the sum or difference of the angles, 

 the one from the other. I shall now investigate a formula that de- 

 termines either angle by itself. 



Let a, (p and d be three angles such, that 



cot 6 = cot(p-coia; then cot = cot a + cot 6. 



Now cot 0-cot « = ^^ _ ?^ = sm{a-<l>) 

 sm (p sin a sin a sin ' 



and cot a + cot = 5^ + £2!i = sinj^+^) 

 sin a sin G sin a sin ' 



Hence, putting for cot and cot a their equivalents - ^ and — - if 



^, , tan tan a' 



appears that 



iftan0 = ?HL°_^. then tan <t> - '"' ° ''" ^ 

 sin(a-0)' ^ sin(a-|-0)" 



Vol. VI. Part I. R 



