94 RELATIONS PERTAIXIXG SIMPLY [BOOK I. 



These transformations, so far as the formulas II. and VII. are concerned, will detain 

 no one, but in respect to formula VI., some explanation will not be superfluous. 

 From the substitution, in the first place, of M (M u) for n, in formula VI., 

 there results 



( } = cos (M u) (cos to sin M sin b sin to cos i cos M sin b -j- sin i cos M cos b ) 

 8iD.(M u) (cos to cosJHfsinJ-|- sm w cos z sin J/ sin 5 smismMcosb). 

 Now we have 



cos w sin M= cos 2 i cos to sin M-\- sin 2 i cos to sin M 

 = sin io cos cos M-\- sin 2 z cos to sin M ; 



whence the former part of that expression is transformed into 



sin i cos (M u) (sin i cos to sin M sin b -\- cos Jf cos b) 

 = sin cos ( Jf M) (cos to sin JV^sin J -)- cos to cos iVcos 5) 

 = cos to sin z cos ( M M) cos (N b). 

 Likewise, 



cos JV= cos 2 to cos JV-\- sin 2 to cos .A 7 &quot; cos (a cos J!f -f- sin to cos sin 3/; 

 whence the latter part of the expression is transformed into 



sm(M M) (cos ^ sin b sin JV cos b) sin (M M) sin (N b). 



The expression VI.* follows directly from this. 



The auxiliary angle M can also be used in the transformation of formula I., 

 which, by the introduction of M, assumes the form 



T*# /^\ _ sino)sin(Jf u) 



Vdr/ ~ A sin M~ 



from the comparison of which with formula I.* is derived 



- R sin (Z t) sin M=. r sin to sin (M 11) ; 



hence also a somewhat more simple form may be given to formula II.*, that is, 

 II.** (^) = - ~ sin (L 1) cotan (M M). 



That formula VI.* may be still further abridged, it is necessary to introduce 

 a new auxiliary angle, which can be done in two ways, that is, either by putting 



