312 PROCEEDINGS OP THE AMERICAN ACADEMY 



projection on a fixed plane containing the axis. Let r denote the per- 

 pendicular let foil upon the axis from dl, and the angle made by r 

 with a perpendicular to the fixed plane. The arc described by a point 

 in dl during the small fraction of a revolution which may be denoted 

 by dO is expressed by rd6; the projection of this arc upon the fixed 

 plane is r cos 6 dO. The projection of the surface, having dl and rdO 

 for its altitude and base, has r cos 6 dO for its base, but dh, not dp, 

 for its altitude. Accordingly, if I denotes the entire length of the gen- 

 erating line, and h its projection on the axis, the ratio of the element, 

 described by the generating line during its movement through dO, 



to its projection on the fixed plane, is expressed by -^ Hence, 



for different values of 6, the projections of the coi-responding elements 

 of the surface are proportional to cos 6. If the projection of each of 

 the surfaces described by the elements of I during its movement through 

 dO is substituted for that surface, and again orthographically projected 

 upon a new plane containing the axis, the final projections of corre- 

 sponding elements of the surface will appear by the same course of 

 reasoning to be proportional to cos 6 cos y, where y denotes the angle 

 made by r with the perpendicular to the new plane. We may regard 

 B as the angle of incidence of parallel rays of light upon a cylinder 

 circumscribing the supposed surfoce, and having the same axis ; y may 

 be regarded as the corresponding angle of emanation. On Lambert's 

 hypothesis, the ratio of light received from the given surface in differ- 

 ent phases will be the ratio of quantities obtained by the integration of 

 cos 6 cos ydO between limits determined for each phase by the extent 

 of the illuminated surface. Denoting the magnitude of the phase, as 

 above, by v, and regarding 6 and y as positive when they are measured 

 from the normal towards the terminator, the limiting values of 

 are — ^ tt at the terminator, and -^ (| tt — v) at the illuminated 

 limb; those of y are i tt — v at the terminator, and | tt at the illu- 

 minated limb. For any value of y, the corresponding value of $ is 

 y — (n — t'). Hence cos 6 cos y d6 = — cos y cos (v -\- y) dy, 

 the indefinite integral of which is — ^ [sin y cos (v -\- y) -\- y cos v]. 

 Integrating between the given limits of y, we have the re<=ult 

 I (sin V — V cos v) ; the constant factor h indicates limitation to half 

 the surface, and vanishes from the ratio between the quantities of 

 light received from different phases. The integration may be some- 

 what simplified by employing the mean of 6 and y as the variable, 

 with the aid of the general formula 



sin (a -f- b) sin (a — h) :^ ^ (cos 2 b — cos 2 a). 



