Photometric Observations of the Sun and Ski/. 271 



whence 



ff ff 



Q m = i a sin I 1 170 dO cos + i a cos I I cZ0 dO sin cos 



JJ JJ 



= 2i a (TT sin + 2 COS a) , (X) . 



29. This is a mere literal result : what is required is (Qz) ; i.e., the 

 lemical action of the hemisphere about Z resolved on the horizontal 



Diane ; that is, the relation between the chemical action in the plane 

 minimum intensity ia (Sun's altitude ) and the " total chemical 

 action of diffused daylight " as observed by Roscoe on horizontally 

 exposed paper. 



The answer (Qz) is identical in form with (Qs) as given in equation 

 (X) above ; but the limits of are functions of which lead to 

 elliptic integrals. 



Referring back, however, to diagram 1, it will be seen that (Qs) 

 liffers from (Qs) by the addition of the gore AYH, the subtraction 

 :>f the gore HYX ; which will be found to be no difference at all ; as 

 the values of the chemical action of each element in the subtracted 



)re are equal to those for a corresponding element in the added 



)re, with the same sign and angles of resolution on the horizontal 

 slane. Hence we must have 



(Q z ) = (Q,) = 2i a (IT sina+2 cos a) (Y). 



Ls this is a result of the first importance, I submit at the end of this 

 >aper in an Appendix, not for publication, the work by which I first 

 rrived at the equation (Y) by laborious transformation of the elliptic 

 integrals, which are reduced finally so that two terms, each irre- 

 lucible by integration in algebraic form, destroy each other. 



30. The results thus arrived at by employing the law of the 

 jsecant are so neat that a suspicion may arise that the law may 



lave been assumed as one lending itself to mathematic manipu- 

 ations. 



I may be permitted, therefore, to state, that the law was arrived 

 it, more than twenty-two years ago, by experiment simply, and the 

 ibject soon after laid aside. The present mathematic investigations 

 only recommenced within the last two years, in order to in- 

 titute a comparison between my old Dacca observations and those 

 Sir H. Roscoe. 



31. In 'Phil. Trans.,' 1870, p. 314, Sir H. Roscoe gives a table 

 showing the total chemical action of diffuse daylight (i.e., of the 



ie sky, the Sun being stopped off) on horizontally exposed paper. 

 These observations were taken in Portugal, with a perfectly clear 

 sky, and I therefore select them for comparison with the foregoing 

 theory and observed values of constants. 



T 3 



