318 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



lation must very nearly hold good for the entire quantity of 

 light (A) which passes through the said distance a. For the 

 case that w— 1 is very small, the investigation may be carried 

 out without much difficulty. Let us imagine a diameter drawn 

 through the sphere in the direction of the rays, and around the 

 end of the diameter let us suppose the entire forward surface 

 divided into narrow zones, and let the attention be fixed on one 

 of those which passes, say the radius i and the width di. The 

 superficial content of the zone, when c denotes the radius of the 

 sphere, is 



= 2cV sin i di ; 



and the projection of this surface upon a plane at right angles 

 to the rays is 



= 2c V sin i cos i di, 



while the projection of the total anterior hemisphere on the same 

 plane is = cV. Assuming the entire quantity of light falling on 

 the sphere as unit, then the quantity which falls upon the ele- 

 mentary zone is 



= 2 sin i cos i di. 



This light meets the spherical surface under the angle i ; and if 

 we denote the corresponding angle of refraction by i', then the 

 quantity of light reflected from the zone, according to the for- 

 mula of Fresnel, is 



„ . . .,.rism^(i-i') ltan2(i-i')l 

 = 2 sm ^ cos idi\- . ^ ). — r/ + - . — s-h — t^ ; 

 L2 sm^ (^ + ^0 2 tan^ [i + t')A ' 



and hence the quantity which enters the sphere is 



„ . . .^. r, lsin2(l-i') ltan2(i-i')-l 

 = '^^^^^"^^^^L^-2 sin^(l + ^0 "2 tan^(^ + ^0 -l' 

 This latter suffers at the posterior surface of the sphere a re- 

 flexion at the same angle, and is therefore a second time en- 

 feebled in the same proportion, so that the quantity of light 

 which passes through the sphere is 



„ . . . ,. r, lsin2(i-i') ltan2 (i-iOT 



= 2 sm e cos e a^ 1 — - . o). — rA — - — s-r^ — 7; 5 



L 2 sm2 (^ + /) 2 tan^ (^ -\-i')J ' 



from which we finally obtain the quantity of light reflected, 



o • • 'A'f^ [^ lsin^(l-iO ltan2(i-i')T\ .,,. 



2sm^cos^6f^ S 1— 1— --v-g-); — t.---- — :rr- — 1\\ r- (H) 



L L 2sm^(l+^') 2tan^(e + /)Jj ^ ' 



