Updegraff — Flexure of Telescopes. 271 



sina cosB = cosb sine — cose sinb cnsA (c) 



sin-4 cos6 = cos^ sinC + cos (7 sin^ cosa, (d) 



we get, 



cos^ sing' = cos^ sina (1) 



cos<5 sin^ = siiiC sina (2) 



sin^ = sin^ cosc — cos<^ sin^ cosa (3) 



cos^ cosg' = sinc^ sin^ + cos<^ cosC cosa (4) 



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



sina s'w(f>= s\nt co^q + cos< sing- sin5 (6) 



sina cos* = co8< sing' + sin^ cosg- sin^. (7) 



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

 have, 



cos^ dd z= — siu<^ sine c^C — coscfy cosC cosa (ZC. 



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



dd = — cosq dZ. (8) 



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



cos^ cos< dt — Qmd sin^ dd = cos> sina dX. 



By (4), 



C08<? dt = s'lnq d%. 



But I = d — a (see Chauvenet's Spher. & Pract. Ast. Vol. 



I., p. 64), and dt = — da, hence, 



cos^ da = — sing' cZ^. (9) 



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



cos^ dd = + COS0 sinC sina da. 

 By(l), 



dd = sing sinC da. (10) 



