OF ARTS AND SCIENCES. 



203 



Reverting to the equations (2G), putting n' i = Mrts, 7r'„ = iV;T„, 

 Ks = /«;?,., where tt, and ;r^ are respectively the equatorial horizontal 

 parallaxes of the sun and Venus ; regarding jtv as the variable, and 

 difFerentiatiug, we get 



dA's = ± »jjj/sin ds drty 

 d^y = -j- -^ cos dv djtv 

 dA'y = ± iV" sin dy ditv 



(49) 



Substituting these values in (48), writing unity for sin ^^, ds for ^„, 

 and d(a for — dp, we obtain 



dg = cos ds{_±. {mM±_ N) sin y tan ds -j- -A^cos y -\- mil/cos ^] (Ztt^ ) 



cos e . r (5<^) 



dco=z— — :'[ ± (mJ/±i\r)cosj' tan (9^ — iV^sinj'-|-?Mil!/'sin^cos(>](^!7r„ } 



From an epheraeris the polar distances of the sun and Venus, and 

 the difference of their right ascensions are taken; and thus two sides 

 and tiie iucluded angle are known in the triangle PSV. The remain- 

 ing parts are given by the formulae 



tan IX = tan A v cos («j — «,,) 

 cos A., 



sin P(, cos cOf, = sin ( A « — u) 



cos fi 



sin Pq sin 0)^ = sin A v sin («s — «,,) 

 sin pg sin a = sin A « sin (a, — «^) 



(51) 



in which we write p^ and co^^ to distinguish the quantities deduced from 

 the ephemeris from the similar quantities q and w obtained from the 

 photographs by means of the equations (30), (32), and (34). 

 Still considering the triangle PSV, we have the relations 



cos Qq = cos A V cos A J -|- sin Ay sin A $ cos («, — a^) 

 sin («, — ciy) cot cOq = sin A s cot Av — cos A j cos (a, — Uv) 



(52) 



Regarding all the parts as variable, differentiating and reducing, 

 we get 



dn^ = sin A„ sin ad(^as -^ «^) -{- cos adAv -j- cos a)„<?As 

 — sinp(,<?(Wo = sin A vCOsad^Uj ^ «^) — sincrc? A v -j- cos^q sin w^c? A $ 



(53) 



