APPENDIX. 299 



76. 



If in the equations of article 60, 



x X= A cosd cos a 

 y Y -= A cosd sin a 

 z Z = A sin 8 



a denoting the right ascension, and 8 the declination, we suppose X, Y, Z known, 



we have 



dx = cos a cosd d A A sin a cost? da A cos a sin$ dd 

 d y = sin a cos d d A -\- A cos a cos 8 da A sin a sin d dd 

 d z = sin d d A -\- A cos d dd. 



Multiply the first of these by sin a , and subtract from it the second multiplied by 

 cos a , and we find 



A cos d d a = dx sin a -j- d y sin a . 



Multiply the first by cos a and add to it the second multiplied by sin a , and 

 we find 



dx cos a -\- dy sin a =. cos d d A A sin (J dd. 



Multiply this equation by sin d and add it to the third of the differential equa 

 tions above multiplied by cos d and we find 



dx cos a sin d dy sin a sin d -\- dz cos d = A dd 



and, therefore, 



d, sin a. , . cos 7 

 d a =. -j-ax -\ -j- dy 



-, ,5, cos a sin 8 -, sin a sin 8 , , cos 3 7 



a o = -. d x -; a y -\ -r- dz. 



d d A 



From the formulas of article 56 of the Appendix are obtained 



dx x dy y dz z 



dr r dr r dr r 



-^ = x cotan (A -f- u) , -^=y cotan (B -f- M) , ^ = s cotan ( C-{- ) 



rfa; . rf . , dz 



- = x sin u cos a , -r-. = r smwcoso, -r-. = r sinwcos c, 

 di d^ di 



and the partial differentials 



dx . dy dz 



r-r ^cose s sins, - = 2 cose, --=xsins 



