90 RELATIONS PERTAINING SIMPLY [BoOK I. 



Multiplying likewise I. by sin (I 8 ), II. by cos (I 8 ) ; and adding together 

 the products, we have 



~~ 



sin u cos i cos (/ & ) cos w sin (I Q ) 



The ambiguity in the determination of u by means of equation IV., is removed 

 by equation III., which shows that u is to be taken between and 180, or be 

 tween 180 and 360 according as the latitude b may be positive or negative ; 

 but if b = 0, equation V. teaches us that we must put u = 180, or u = 0, accord 

 ing as sin (L 1) and sin (I 8 ) have the same or different signs. 



The numerical computation of the formulas l\ r . and V. may be abbreviated in 

 various ways by the introduction of auxiliary angles. For example, putting 



i if IAJS \ J^ Aft / A 



- = tan^4. 



sin (L /) 



we have 



_ sin A tan (L & ) 



sin (A -\- i) 

 putting 



tan i sin (L /) 



,, ._ . = tan B, 

 cos(L-Q) 



we have 



cos B sin b tan (L Q ) 



tanz&amp;lt; = - , ,, i , . . 



sin (/-p 6) cost 



In the same manner the equation V. obtains a neater form by the introduction 

 of the angle, the tangent of which is equal to 



., tan(/ Q) 



cos z tan it, or - ~^- . 

 cost 



Just as we have obtained formula V. by the combination of L, IE., so by a combina 

 tion of the equations II, III, we arrive at the following : 



r 







sin u (cos t sin i sin (/ Q ) cotan b) 



and in the same manner, by the combination of equations L, HI., at this ; 



Q) 



T 



cos u sin u sin t cos (I Q, ) cotan b 



