92 RELATIONS PERTAINING SIMPLY [BOOK I. 



the earth, which of course can have no sensible effect in differential formulas. 

 The following formulas accordingly are at hand, in which, for the sake of brevity, 

 we write w instead of I 8, and also, as above, A in the place of A cos b. 



A cos to = r cos u R cos (L 8 ) = 

 A sin a) = r cos ismu R sin (L & ) = i\ 

 /I tan b=.r sin i sin u = ; 

 from the differentiation of which result 



cos w.d A // sin w.deo = d 



sin w.d^/ -\- A cos co . d o&amp;gt; = d?j 



. -, . 



1 cos 6 



Hence by elimination, 



I _ _ sin to . d | -f- cos to . fl i; 



A 



-, , _ cos co. sin 5. dj sin to sini.d)y-(- cosi.d 



A 



If in these formulas, instead of , 77, t, their values are substituted, do* 

 and d$ will appear represented by dr, dw, d/, dQ; after this, on account of 

 d/=doj-|-d&, the partial differentials of I and b will be as follows : 



I. A ( } = sin w cos u -(- cos w sin M cos i 



-rr ^ / dz \ 



II. : I =- 1 = sin w sm ; + cos w cos u cos z 



r VdM/ 



TTT ^ /dZ\ 



ILL T-. ) = 



r \d/ 



cos CD sin M sin i 



V. ^(^) = cos w COSM sin b smto sin u cos i sin b -f- sin M sin i cos i 



, rT ^/di\ . . 



V 1. ( -j- I = cos w sm M sin o sin (a cos M cos i sin o -}- cos u sin z cos 



irrr ^ /&amp;lt;! *\ .... 



V ll. IT-. I = sm w sin M sin i sin o -j- sm u cos z cos b 



VIII. -^-Q ^| = sin b sin (Z 8 w) = sin b sin (Z /) . 



