

254 



liutherfurd Photographic Measures. 



If we do so, we shall have, omitting only certain very minute 

 terms of the third order :* 



(b) 



g — s = r cos I — r' cos V y 



s (X — l) = — r sin I -J- r' sin V ) 



in which a and X are the values of s and I after correction for re- 

 fraction, and r and r' the vertical refractions of 8 and S'. The 

 details of the rigorous deduction of equations (b) are omitted here, 

 for the sake of brevity. But the equations as they stand can easily 

 be obtained by an inspection of the figure. 



Equations (b) are extremely accurate, as well as simple in form ; 

 but they are inconvenient for practical purposes since they involve 

 r f and V ', quantities which are different for every star on the plate. 

 We shall therefore find expressions for r' and V in a more conve- 

 nient form, and at the same time introduce the quantities s and z 

 into the second members of the equations. The expansions will be 

 carried to terms in s 2 inclusive. Following Bessel, we put: 

 r — k tan z , r' =k' tan z' , 



where k and k' are the usual refraction quantities with the argu- 

 ment "apparent zenith distance." We may also write: 



and therefore: 



/ dk 



k = — (z f — z) 

 dz 



dk 



r' = k tan z f -\ tan z' (z' 



dz 



o 



Moreover : 



tan z f = tan z -f- (z f — z) sec 2 z -\- (z f — z) 2 tan z sec 2 z . 

 cos V = cos I — (V — 1) sin I — ^(V — I) 2 cos I . . . . 

 sin V = sin I -f (/' — I) cos I — ±(l' — l) 2 sin I . . . . 



We also have, to terms of the second order, inclusive :f 

 z' — %i= — s cos I -j- J s 2 sin 2 1 cot z 



I = s sin I cot z -\- -J- s 2 sin I cos I (i -\- 2 cot 2 z) 



* While the present investigation was being printed, Professor Newcomb 

 called my attention to an investigation of differential refraction, in which he 

 has used fundamental formulae somewhat similar to equations (b). I was not 

 previously aware of this work of Professor Newcomb's, which was published 

 in his Report on the Transit of Venus, Bee. 8-9, 1874 (Ex. Doc, U. S. Senate, 



1879). 



f For a demonstration of these formulae, see Jordan, Handbuch der Vermess- 

 ungskunde, dritte Auflage, 1890, vol. iii, p. 313. 



