OF ARTS AND SCIENCES. 31 



For any variation whatever of the angle i we have : 



^i = T-- FT N = 



lo cos (r, — - T-J COS I 



and hence, from transits over the line x^, 



2) = 15 cos 8 (t2 — Tj) sin {i -\- A^) 



= 15 cos 8 (r, — Tj) sin i -I- — — r A 



^ - ' L locos (T2 Tj) COS ?J 



and from the transits over the line x.-^, 



Z> = 15 cos 8 (t, — To) sin i — -^ t—, r ; 



^ ■* •^^ L lo cos 6 (r^ — Tg) cos ij 



If therefore the times of transit of each limb are taken over the 

 lines a^j and x.^, any error in D due to an erroneously assumed position 

 angle will be eliminated. 



It must be noted, however, that any error in A^ arising from an un- 

 known error in the angles between a-^, x^, and x^ will be only partially 

 eliminated. Designating by i and i' the angles which x^ and x^ make 

 with X,,, and their variations on account of errors of graduation by Ai 

 and Az' respectively, we have, from transits over Xj, 



D ■= \5 cos 8 (t^ — Tj) [sin i -\- cos i A^] 

 and from the transits over x^-, 



D = 15 cos 8 (T^ — Tg) [sin ^'-{- cos i' At'] 



or, since i' = 180° — i nearly, 



Z> = 15 cos 8 (r^ — Tg) [sin z — cos i Ai'] 

 whence 



X> = -V- cos 8 [[(r, — rj + (r, — r^)] sin ^ 



-j- cos I [(t2 Ti) At* (t^ T.) Aj']] 



The only case, therefore, in which the elimination will take place is 

 that iu which 



(r, — tJ M = (t, — T3) A{' 



But since, on Dec. 6, the time required for Venus to make a complete 

 transit over a line having i = 20° was only 24', the effect of any 

 small error in the graduation will be practically insensible. 

 For the equatorial diameter we have : 



D= 15 cos 8 (t2 — Tj) sin (90° + Ai) 

 Unless Az, therefore, is very large, we shall have : 



Z) = 15 cos d (Xg Tj) 



Assuming the same constant of differential refraction for Venus 

 north and for Venus south, any error in the observed value of Z> 



