204 PROCEEDINGS OF THE AMERICAN ACADEMY 



If now «., «^, As, Avi'^si and Tt^ are the tabular values of the 

 right ascensions, polar distance-*, and parallaxes of the sun and Venus ; 

 and if as-\-d<is, «r -|- rf«;,, As -\- dAs, Av -\- dAv, rts-^drts, 

 nv -f- drtv, are the true values of the same quantities; then we must 

 have 



Q -\- di) = Q^ -\- dn^ ) 



[ (54) 



(o -\- d(o =^ cOq -\- c?a)„ ) 



Substituting the values of dQ, dQ^, doo, and dco^ from (50) and (53), we 

 obtain finally 



( Po — P ~f~ ^'" ^i> sin ""^(tti ~' Km) -1- cos (rdAy -\- cos w.dA. ) 



= ] [ (55) 



( — cos e^ [J; {mM -i- iV) sin y tan ©^ + iV cos 7 + ^^^ cos fijdir^, ) 



(<l)^) — w : — - [sin A^ cos irdla,'^ay) — sin <rc?A„-j-cos p^ sin uodA^] ) 



^= cose ''"''' (56) 



V ; — -^ [-l-(/ftJ/-J-.iV)cos 7 tan tf^. — iVsin7-|-'«-3i'siny3cosp]c?iry ) 



siu p 



Each photograph furnishes one conditional equation of the form (55), 

 and another of the form (56), and from all the equations thus obtained 

 the values of 6?((iCj ~. «^), dAs, dAv-> and rf;Tr^are found by the method 

 of least squares, the resulting value of the solar parallax being 



ns -\- m drty (57) 



At the time of the last transit the value of m was 0.2684. Tho 

 term ± {mM ± N) of equations (55) and (56) is to be interpreted 

 thus : When the sun and Venus are on opposite sides of the meridian, 

 it will be -\- {mM-\- N) ; when the sun and Venus are on the same 

 side of the meridian, if the sun is most distant from the meridian, it 

 will be -|~ (juM — N) ; but if Venus is most distant from the me- 

 ridian, then it will be — {mM — N). 



It will not escape notice that those parts of equations (55) and (56) 

 which correspond to q^ -\- do^, and w^ -\- doj^, of the equations (54), 

 are general for the whole earth, and can therefore be tabulated at 

 suitable intt-rvals for the period of the transit ; while the terms which 

 correspond to p -f" ^Qj and w -|- da, must be computed specially for 

 each photograph. 



Washington, Nov. 15, 1876 



