128 RELATIONS BETWEEN SEVERAL [BOOK I. 



By transformations precisely similar, the development of which we leave to the 

 skilful reader, are found 



[29] cos(/ g] tan 2 &amp;lt;o = sin. (F-\- G) sin 



When the first members of these four equations are known, J (F G] and 



will be determined from 27 and 29 ; and also, from 29 and 30, in the same manner, 

 and 



aa 



the doubt in the determination of the angles i (F G), $ (J?-\-G), is to be so 

 decided that P and Q may have the same sign as sin g. Then \ (f and 



will be derived from P and Q. From R can be deduced 



and also 



sin 2 /Vr 1* 

 - 



unless we prefer to use the former quantity, which must be 



+ y/ (2 (1+ sin 2 Iff) cos/) = + y/ ( 2 (L sin 2 * ff ) cos/), 

 for a proof of the computation chiefly, in which case a and p are most conven 

 iently determined by the formulas 



, sin/Vr/ b 7 



= v^ , a= -- , p = ocosq&amp;gt;. 



sing cos gr * 



Several of the equations of articles 88 and 95 can be employed for proving the 

 calculation, to which we further add the following : 



in 2 co / ri . .~ . 



* / = e sin G sin a 



s 2o) V aa y 



2 tan 2 o) 

 cos 



