316 PROCEEDINGS OF THE AMERICAN ACADEMY 



illuminated and visible, the liglu received from the visible parts of 

 slopes facing towards the terminator is equal to that received from the 

 continuous surface of the same region of the smooth cylinder when the 

 jjhase is diminished by an amount equal to the angle of elevation. In 

 this case, the formula obtained by Zollner is partially api)licable, since 

 it is accordant with the fact that no light will be received from the 

 furrowed cylinder until the phase is larger, by an amount equal to the 

 angle of elevation, than the initial phase of the smooth cylinder. As 

 this was the only point considered iu ZoUner's geometrical interpreta- 

 tion of his formula, the i-esult of his analysis appeared to be confirmed. 

 In considering the quantities of light rect:ived from the slopes facing 

 the illuminated limb, it will be convenient to assume what is approxi- 

 mately true iu the case of the Moon, that during the changes of phase 

 the angles of emanation at given points of the illuminated surface are 

 constant. Upon Lambert's hypothesis, equal quantities of light will 

 emanate from lines equally inclined to the line of sight, but of different 

 lengths, provided that these lines have received equal quantities or 

 incident light. In the ligure, the lines CA, CE, receive equal quanti- 

 ties of incident light, and if we now suppose the phase diminished by 

 the revolution of CT, and the lines parallel to it, about C as a centre, 

 through the angle ACB, the observed light from CE will be equal 

 to the light formerly received from CA. Since, however, we have 

 assumed the direction of CT with respect to CA to be invariable, this 

 result signifies that, when the phase is diminished, a quantity of light 

 equal to tliat formerly received from CA is now received, not from 

 CE, but from an equal arc of the circumference of which CE is part, 

 situated as far from the new terminator as CE was from the original 

 terminator. It may also be observed that BF -|- CE > BC. Ilcnce, 

 in order to assume, with Zollner, that the light received from the 

 smooth cylinder in the reduced phase is equal to that previously re- 

 ceived from the furrowed cylinder, we must assume at the same time 

 an increase of reflecting power in i)art of the smooth cylinder. Leav- 

 ing other geometrical considerations to suggest themselves, we may 

 now construct the formula required to represent the phases of the 

 i'urrowed cylinder. 



Returning to the figure, let s denote the length of tlie slope AB or 

 AC; /?, the angle of elevation ABC or ACB; tt — v, the supplement 

 SBT' of the magnitude of the phase ; $, the angle of incidence on EC, 

 of which MEB is the complement; and y, the corresponding angle of 

 emanation, the complement of which is NFC. As before, 6 and y will 

 be regarded as positive when measured from the normal to BC towards 



