Updegraff — Flexure of Telescopes. 271 



sina cosi? = cos6 sine — cose sin6 cos^4 (c) 



sin^4 cos6 = cosi? sinC -f cosC sini? cosa, (d) 



we get, 



coso" sing = cos<£ sina (1) 



cosa" sin£ = sin? sina (2) 



sino" = sin<£ cos? — cos<£ sin? cosa (3) 



cosa* cosg = sin<£ sin? + cos$ cos? cosa (4) 



eosd cos£ = cos<£ cos? + sin$ sin? cosa (5) 



sina sin<£ = sin£ cosa + cos£ sina sinJ (6) 



sina cos? = cos£ sina + sin£ cosa sin*?. (7) 



Differentiating Eq. (3) regarding d and ? as variable, we 

 have, 



cosd dd = — sin<£ sin? dZ — cos</> cos? cosa dt. 



This by means of Eq. (4) reduces to, 



dd = — cosa d:. ( 8 ) 



Differentiating (2) regarding d, t and ? as variable, we have, 



cosd cos£ dt — sind sin< dd = cos? sina cZ?. 



By (4), 



cos<? dt = sina d?. 



But t — 6 — a (see Chauvenet's Spher. & Pract. Ast. Vol. 

 I., p. 64), and dt — — da, hence, 



cosd da = — sing a 7 ?. (9) 



Differentiating (3), regarding d and a as variable, we have, 



coso d3 — -{- cos<£ sin? sina da. 

 By (1), 



dd = sing sin? da. (10) 



