200 PROCEEDINGS OF THE AMERICAN ACADEMY 



In the preceding development of the theory of the horizontal 

 photoheliograph, continual reference has been made to the centres 

 of the sun and Venus, but of course it will be understood that all the 

 equations apply equally well to any other pair of celestial objects 

 which may have been photographed with the same apjjaratus. 



As the horizontal photoheliograph was much used in observing the 

 last transit of Venus, it is perhaps desirable to give here a direct 

 method of deducing the solar parallax from the photographs then 

 obtained. For that purpose consider the quadrilateral PZS'8, which 

 is composed of the triangles PZS' and PS'S. In the triangle PZS' 

 we have the relation 



sin A's cot B =z sin ^'g cot <f' -(- cos ^', cos A' 3 (36) 



in "which qp' is the colatitude, ZP; and £ is the angle ZS'P. Con- 

 sidering all the parts, except gj', as variable, and differentiating, we get 



dB = — - cos C'i (cot flp' c?l's + sin A's dA's) 



sill A'g 



4- cos A's ( - — -- sin ^'s d^'s — 4^^ cos B dA's) 

 \sm A'g sin A'g / 



(37) 



To obtain approximately the maximum value of this differential, 



we remark that A's can never exceed ± 13' ; and as sin B must 



, IT 1 • ^, sin S , , , . , sin'^ B 



always be less than sin A's , must be less than unity ; and 



sin A\ •' sin A'^ 



must be less than sin A's , that is, it must be less than 0.004. ^'s can 



never differ from 90° by more than ± 13', and therefore its sine may 



be taken as unity, and its cosine cannot exceed 0.004. If the latitude 



of the place of observation is less than 50°, g;' will be greater than 



40°, and its cotangent will be less than 1.20. Substituting these 



values in the second member of equation (37), all the terms except 



the last become evanescent, and we may write, without an error of 



one part in ten thousand, 



dB= — dA', (38) 



But the only way in which A's can be made to vary is by varying the 

 adopted value of the solar paralkax, tts. Hence, as ZS' is nearly 90°, 

 dA's is the resolved value of drts, and as it can never exceed that 

 quantity, it is safe to write 



dB = — drts (39) 



