OP ARTS AND SCIENCES. 813 



To find the ratio of light, for a given phase, between two different 

 surfaces of revolution, we have only to determine the correspondino- 

 ratio of light for single elements of each surface, with the same values 

 of 6 and of y for each element ; the values to be assumed in practice are 

 ^ = y = 0. In the case of the sphere and the circumscribing cylinder, 

 the height of which is equal to the diameter of the sphere, let ^ denote 

 the angle between the axis and the radius drawn to any point in that 

 element of the sphere for which 6 and y vanish. At this point, the 

 angles of incidence and emanation are each equal to the complement 

 of ^; and the quantity of light received from the immediate vicinity of 

 the point will have the ratio sin- ^ to that received from an equal sur- 

 face at any point of the corresponding element of the cylinder. But 

 as the width of the spherical element, at the given point, is the pro- 

 duct by sin ^ of the width of the cylindrical element, the quantity of 

 light received from the element of the sphere, at the given point, will 

 have the ratio sin-^ ^ to that received from an equal length of the ele- 

 ment of the cylinder. Accordingly, if the radius of the sphere is the 

 unit of length, the ratio of the quantities of light received from the 



spherical and cylindrical elements is )j I sin^ 'CdC Changing the 



variable to — cos ^, which we may represent by x, we have 



1 Tsin' I dt = I rtl — ^') dx = 



the ratio found by Zollner (p. 40), from an independent determina- 

 tion of the phases of the cylinder, combined with Lambert's similar 

 proposition relating to the sphere. It is also sufficiently apparent, 

 although not distinctly set forth, in Seidel's statement of Lambert's 

 demonstration, that this ratio exists. 



If, instead of the sphere, we consider a right cone, inscribed in a 

 cylinder of .the same height and base, the coefficient required to reduce 

 the phases of the cylinder to those of the cone is readily perceived to 

 be J cos ;)(, in which ;^ denotes the inclination of the generating straight 

 line to the axis. The direct determination of the phases of the cone 

 is nearly as simple. The cosines of the angles of incidence and ema- 

 nation will be respectively expressed by cos ;i( cos $ and cos x cos y ; 

 the slant height of the cone is proportional to sec ^ ; and the ex- 

 pression to be integrated becomes 



L 



I cos X cos y cos (v -\- y) dy. 



