320 PROCEEDINGS OP THE AMERICAN ACADEMY 



by Seidel,* equation (6) is somewhat better adapted to numerical com- 

 putation than equation (5), Equation (2) may be very readily com- 

 puted from Seidel's table. 



By means of equations (2) and (6), numerical expressions may be 

 obtained for the quantities of light corresponding to the phases of the 

 furrowed cylinder for assumed values of yS, in terms of the quantity of 

 light received from the smooth cylinder in opposition. If we adopt as 

 the unit the quantity of light received from the furrowed cylinder 

 itself when v = tt, we must divide each result by the corresponding 



value of the expression - sec /? (tt — 2/3 sin- /S). 



The differential coefficient of the value of L in equation (6), taken 

 with respect to v, is 



- sec ;8 [v sin v — 2/3 sin /3 cos (v — /S)]. 



The value of L, accordingly, if /3 > 0, does not reach a maximum 

 when V = TT. For this value of v, L is greatest when the value of /3 



is about 38° ; when (3 = 60°, L = 1 ; and for the limiting value 



2 

 ft =■ 90°, L = -. This last result appears from equation (3) ; equa- 

 tion (6) is not applicable to the case without previous reduction to (3), 

 since v cannot exceed 2 /3 when /? = ^- tt. 



Numerical values of L for six assumed values of /3 are given in 

 Table I. The unit of light, in this part of the table, is the quantity 

 received from the smooth cylinder when ?; = tt. The last three col- 

 umns contain results derived from the work of Zollner. The first of 

 these columns contains the values of L •according to his formula, when 

 (3 = 52°, and when the amount of light received upon this supposition 

 is regarded as unity if v = tt. The next two columns relate to the 

 twenty-two sets of observations, the results of which are given on 

 page 102 of his work. These results are here arranged in accordance 

 with the values of v, for which, the phase being always large, Zollner 

 substitutes the corresponding elongations of the Moon. The values of 

 V are here printed in Italics when the observations were made after full 

 moon. The observation made when v = 179° is assumed by Zollner 

 to be in accordance with his formula, in order to compare the other 

 observations with the results of the theory. This assumption is here 

 retained. 



* Untersuchnngen iiber die Lichtstarke der Planeten Venus, jNIars, Jupiter 

 und Saturn. Miinclien, 1869. (Pp. 100-102 contain the table above mentioned.) 



