OP ARTS AND SCIENCES. 319 



When V = 2 (3, this result coincides, as it should, with that from 

 I [sin (v-l3)-~{v- (S) cos (v - /3)]. 



The collection of the separate results of the inquiry furnishes the 

 following rule for computing, upon Lambert's hypothesis, the phases 

 of a right cylinder, with a surface occupied by regular furrows, parallel 

 to the axis, and of dimensions too small to be comparable to those of 

 the cylinder, which is supposed to be placed at a great and constant 

 distance from the source of light and also from the observer, with 

 its axis perpendicular to the rays of incident light and to the line of 

 sight : — 



If we adopt as the unit of measurement the quantity of light received 

 from a smooth cylinder equal in height and diameter to the furrowed 

 cylinder, when the magnitude of the phase is an entire semi-circum- 

 ference, denoting this magnitude, for any phase, by v ; the angle of 

 elevation, formed by the base and slope of each ridge, by /3 ; and the 

 observed quantity of light, by L ; then, if /5 = 0, 



L = - (sin V — V cos v). (1) 



If /? > 0, and y < ^, L = 0. 

 lfv>(3>0,andv<2/3, 



L = - [sin (v—(3)—(v — (3) cos {v — yS;]. (2) 



Ifv = 2/?, 



L = -(sin/3 — /3cos/3). (3) 



If V > 2 /5 > 0, 



L = - [sin i3cos(v — 2 13) — ft cos (v — /5)] 



_]_ Jl sec /3 [cos 2 /3 sin (v — 2 (3) — {v — 2 (3) cos v^, (4) 



IT 



By the collection of the terms of equation (4) which contain re- 

 spectively sin V and cos v, the equation is reduced to 



L = - sec /S [(1 — /J sin 2 j8) sin y — (t; — 2 /? sin^ yS) cos v]. (5) 



Equation (5) is readily transformed to 



L = - sec ^ [sin w — v cos v — 2 /? sin /? sin (v — /?)]. (6) 



TT 



Since the expression log ^ (sin v — v cos v) has been tabulated 



