OP ARTS AND SCIENCES. 201 



In the triangle PSS' we have the relations 

 cos %-=■ — cos T/; cos (1^ ~ <'$) -\- sin i/; sin (<, ~ t' ^ cos A j 

 cos T/^ = — cos '1^ cos (i's ~ t'^ -j- sin ;f sin (4 ~ i'^) cos A '« 



(40) 



Considering all the parts as variable, differentiating, and reducing, we 

 get 



di = sin A 's sin (ts ~ t\) dAs — cos A ', d(t^ ~ t'^) — cos S8'd\p) 



dip = sin A s sin {t^ ~ t't) dA's — cos A ^ dQ^ ~ t'j) — cos SS' dx ) 



Adding, this becomes 



^~^ ^— l + uos SS' ^ ^ 



To obtain approximately the maximum value of this differential, 

 we remark tliat at the time of the transit the sun's north polar distance 

 was 112° 49', and therefore sin As = 0.922, and cos As = 0.388. 

 If the latitude of the place of observation is not greater than 50°, the 

 value of A's will lie between 130° and 180°; and consequently its 

 sine will not exceed 0.766, and its cosine cannot be gi'eater than 

 unity. Sin (tg ~ i's) cannot exceed unity. Further, as the triangle 

 PSS' can only be varied by varying the assumed value of the solar 

 parallax,* of As and dA's are the resolved values, and d {tg ~ t'g) is 

 the sum of two resolved values of dTts. It is therefore certain that 

 «?As and </a's are not greater than dTts, and that d{ts^t's) is not 

 greater than 2 drCs. Substituting these values in equation (42), and 

 adding all the terms, without regard to sign, we get 



dy^dip= ^^'^''' (43) 



^ ~ ^ 1 + C08 SS' ^ ^ 



* Strictly speaking, although the point S' can only be varied by varying the 

 assumed value of the solar parallax, the point S can be varied, not only in that 

 way, but also by varying the tabular place of the sun. In practice it will 

 probably be best to neglect at first the errors of the solar tables, and after- 

 wards, when tiiey become known from the solution of the final equations, to 

 compute rigorously the value of dx + d\l/ for each photograph, by means of 

 (42), and in all cases where it exceeds two or three seconds, which will rarely 

 'happen, the corresponding conditional equations of tlie form of (55) and (56) 

 may be corrected so as to accord with the new values of the solar elements, 

 and a second solution will give very accurate results. 



