OP ARTS AND SCIENCES. 311 



a small surface, at a great distance from the source of liglit, will be 

 proportional to the magnitude of its orthographic projection upon a 

 plane perpendicular to the incident rays ; that is, if the surface is plane, 

 to the cosine of the angle of incidence. Its apparent illumination, 

 then, upon Lambert's theory, will be proportional, at any single point, 

 to the product of the cosines of the angles of incidence and emanation 

 at that point, if we understand the angle of emanation to signify the 

 angle between the normal to the illuminated surface and its direction 

 from the observer. This theory assumes that the reflection of light 

 from the given surface takes place irregularly, specular reflection being 

 altogether neglected. For this reason, as well as for others stated by 

 Zollner (p. 24), Lambert's formula seems likely to be fully apiilicable 

 only to somewhat translucent objects. A strictly opaque object, if 

 smooth, will probably exhibit the phenomena of specular reflection to 

 such an extent as to destroy the value of the formula ; if rough, the 

 theoretical quantity of irregularly reflected light will be variously 

 reduced in different phases by the shadows of the prominences. 



With the assumptions above explained, Lambert arrived at the 

 result that the light received by irregular reflection from a distant 

 sphere would vary with the phases of the sphere in accordance with the 

 expression sin v — 1> cos v, where v denotes the angular magnitude of 

 the phase. It may also be deflned as the exterior angle, at the illumi- 

 nated object, of the triangle formed by the straight lines connecting 

 that object, the source of light, and the observer. If the total quantity 

 of light received from the sphere when in exact opposition is regarded 

 as unity, the quantity in any other phase will accordingly be denoted by 



- (sin V — V cos v), provided that we regard as constant quantities 



TT 



the intensity and magnitude of the source of light, the magnitude and 

 reflecting power of the sjihere which it illuminates, and the distance of 

 the sphere both from the source of light and from the observer. The 

 coefficients depending upon these quantities will accordingly be neg- 

 lected in the present inquiry. 



Zollner, in order to simplify the discussion of the lunar phases, 

 shows that the variable factor sin v — v cos v will be applicable, on 

 Lambert's hypothesis, to the phases of a right cylinder, with its axis 

 perpendicular to the plane of the triangle having its vertices at the 

 source of light, tlie cylinder, and the observer. The proposition may 

 apparently be extended to any surface of revolution subjected to the 

 same condition. Let dl denote an element of the generating line, 

 dh its orthographic projection on the axis, and dp its orthographic 



