Updegraff — Flexure of Telescopes. 271 



sina cos^ = cos6 sine — coac sin6 co3^ (c) 



sin^ cos6 = co3^ sinC + cosC &mB cosa, (d) 



we get, s 



cos^ sing = 008^ sina ( 1 ) 



cos^ sin^ = siiiC siua (2) 



hmd = sin^ cosC — cos<^ sinr cosa (3) 



cos^ cosg = s>in</> sinC -f- co8<^ cosC cosa (4) 



cos5 cos< = cos<^ cosC + sin^ sinC cosa (5) 



sina sin<^ = sin^ cosg + cos< sing sinJ (6) 



sina cosC = co8< sing- + sin< cosg sin^. (7) 



Differentiating Eq. (3) regarding ^ and C as variable, we 

 have, 



cos^ dd = — 8in<^ sinC dZ — cos<^ cosC cosa dC- 



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



dd = — cosj dZ. (8) 



Differentiating (2) regarding <J, t and c as variable, we have, 



cos^ cos< dt — sin^ sin< dd = cosC sina dz. 



By (4), 



008^ dl — sing- d^. 



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



I., p. 64), and di = — da, hence, 



cos<? da = — sin^' dZ. (9) 



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



cos(5 do = + cos</> sinC sina da. 



By (1), 



dd = s'mq sinC da. (10) 



