314 PROCEEDINGS OF THE AMERICAN ACADEMY 



For a frustum the height of which is insignificant in comparison with 

 the total height of the cone, the coefficient h disa2)pears. 



Zollner regarded the analogy existing between the phases of a 

 smooth cylinder and a smooth sphere as an indication that the effect 

 of rouo'heniun' the surfaces would be somewhat similar in the two cases 

 (p. 51). Although it is doubtful if this extension of the analogy can 

 be carried far, it is still reasonable to think that it may be of use in 

 obtainiuij some notion of the kind of modification which Lambert's 

 formula will require when it is applied to the j)hases of a mountainous 

 body like the Moon. Zijllncr accordingly considered the consequences 

 of covering the surfaces of the cylinder with furrows parallel to the 

 axis, and obtained a general expression for the phases of this furrowed 

 cylinder without regarding the dimensions of the furrows as insignifi- 

 cant with respect to the diameter of the cylinder. By so regarding 

 them, the expression was afterwards reduced to 



sin {v — 13) — {v — /5) cos (v — /3), 



if we omit, as before, various f;ictors independent of the phase. In this 

 expression, /3 signifies the angle between the base and the slope of 

 each ridge, called by Zollner the angle of elevation. lie found his 

 own photometric observations between half and full moon well satisfied 

 by this formula when the value of /3 was assumed to be 52°. 

 The geometrical significance of the expression 



sin {v — /?) — (y — ft) cos (v — ft) 



was partially developed by Zollner (p. GG). A sufficient extension 

 of the process would liave enabled him to correct an error in his 

 analysis which, as it happened, did not attract his attention. The 

 expressions for the elementary surfaces F^ and Fg (pp. 55 and 56) 

 are not applicable throughout a sufficiently large phase. Both the 

 general expression and the simplified formula above mentioned are 

 therefore incomplete. The correction of the general expression is 

 unnecessary for our present purpose. "NVe may accordingly regard a 

 section of the cylinder perpendicular to its axis as a regularly serrated 

 circle, each serration being so small that the arc forming its base must 

 be considered as a straight line. 



In the following figure, let A represent the junction of two adjacent 

 serrations, the vertices of which are at B and C. From B and C 

 draw the parallels BS, CS', directed towards the source of light, and 

 BT', CT, directed towards the observer. From A draw AM parallel 



