RELATIONS BETWEEN SEVERAL [BOOK I. 



[10*] fl = = iC^ 



[12*] M= - (L - sin* | ,) + (L - sin 



in which the doubtful sign is to be determined in the same manner as before. 



89. 



We have now two things to accomplish ; first, to derive the unknown quan 

 tity g as conveniently as possible from the transcendental equation 12, since it 

 does not admit of a direct solution ; second, to deduce the elements themselves 

 from the angle g thus found. Before we proceed to these, we will obtain 

 a certain transformation, by the help of which the computation of the auxiliary 

 quantity I or L is more expeditiously performed, and also several formulas after 

 wards to be developed are reduced to a more elegant form. 



By introducing the auxiliary angle w, to be determined by means of the 

 formula 



we have 



J !L 4. JL. 2 _|_ (tan (45 + w) cotan (45 -f w)) 2 = 2 -f 4 tan 2 2 o&amp;gt;; 



whence are obtained 



, _ sin 2 !/ , tan 2 2o ,- _ ^sin 2 ^/ _ tan 2 2 &amp;lt;u 



cosy cosy : cosy cosy 



90. 



We will consider, in the first place, the case in which a value of g not very 

 great, is obtained from the solution of the equation 12, so that 



may be developed in a series arranged according to the powers of sin \ g. The 

 numerator of this expression, which we shall denote by X, becomes 



^ sin 8 kg V- sin 5 i y | sin 7 i g etc. ; 



