OP ABTS AND SCIENCES. 



197 



Tlie next step will be to free these zenith distances and azimuths 

 from the effects of refraction and parallax. For that pur[)ose, con- 

 sider the spherical triangle ZS S'*, Fig. 4 ; but instead of limiting the 

 points S, S,j, S', S'^, to the sun, let them represent any heavenly body 

 whatever. Then, ZS' = l' ; ZS'^ = ^" ; S'S'^ = SS^j^ = r — n' ; 

 S'ZS'^it = A" ~ A' ; and ZS^^S' = d, the value of which is given by 

 equation (9). The relations subsisting among these parts are, 



sin ^ cos (A" ~ A') = 



tan m = tan (r — n') cos d 



cos (;• — it') sin {C" — "0 



cos III 



sin ^' sin (A" ^ A') = sia (r — n') sin d 



(23) 



To simplify these equations we remark that (r — n') will rarely 

 amount to 5', and as cos d must always be less than unity, we may 

 write with all needful accuracy 



m = (r — 7i') cos d 



(24) 



^ will never differ from 90° by so much as 30', and therefore its sine 

 may be taken as unity ; while as {A" ~ A') can scarcely amount to 

 5', and will usually be far less, we may write unity for its cosine, and 

 substitute the arc for its sine. We thus find 



C' = C" — (r — 7t') cos d 

 A' = A" q: (r — 7t') sin 6 



(2o) 



As these equations are perfectly general, we have only to substitute 

 in them, for ^" and A", the apparent zenith distances and azimuths of 

 the images of the sun and Veuus, given by the equations (22), and 

 there results the true zenith distances and azimuths of the images of 

 these bodies, which are 



Ts = Co — 5C« — (^s Tt's) cos ds 



A's = ^0 + ^^s =F (r-s — n's) sin ds 



^^ = To — 8Xv — (?V — Tt'v) COS dy 

 A'v = Jq -[- 8A„ =F (r^ — n'v) sin d^ _ 



(26) 



in which the upper signs are to be taken when the body is west, and 

 the lower when it is east, of the meridian. Strictly speaking, the 

 value of d will not be the same for Venus as for the sun, but the 



