318 PROCEEDINGS OF THE AMERICAN ACADEMY 



and the second in Fig. 3. The length of BD, the visible portion of 

 the slope AB, is readily found to be ■ —-^ — : dy ; that of CD, the 



^ ' ■^ cos (y — ^) ' 



COS Q 

 ■ illuminated portion of AC, is - — dy. The expressions to be 



integrated in the two cases are respectively cos y cos (6 — /?) dy, 

 and cos 9 cos (y -j- /?) dy. The first may be reduced to 



— cos y cos \_(v — p) -\- y'\ dy, 

 the second to 



cos (v -\- y) cos (y -\- jS) dy, 



and this, again, to 



-cos (y + ^) cos[(^;-;8) + (y + ^)]^(y + y9). 



To determine the limits of integration, we are to consider that while 

 ?; < /? no illuminated portion of the furrowed cylinder can be visible ; 

 and also that for values of v between /? and 2 /? two illuminated por- 

 tions of the cylinder will be visible, separated by a dark interval. 

 Part of each slope facing towards the terminator will be illuminated, 

 and will include a visible portion, for values of y extending from i tt 

 to I TT — {v — yS). These are accordingly the limiting values of y 

 in the integration of — cos y cos [ (y — /^) "j- y] dy. Part of each 

 slope facing towards the illuminated limb will be visible, and will 

 include an illuminated portion, for values of y extending from -^ tt — y8 

 to ^ 7r — V, where the terminator is reached. The expression 



- cos (y + /?) cos [(. _ ^) + (y + ^)] (f (y + /5) 



is consequently to be integrated between the limits 



^ IT and I TT — (v — /3). 



It is only necessary to compare these expressions, and their limits, 

 with the results obtained above (p. 312), to see that each of the defi- 

 nite integrals required will have the value 



1 [sin {v — IB)-(v—P) cos {v — fi)] ; 



it is also apparent from geometrical considerations that this should be 

 the result for values of v between (i and 2 /?. When v > 2 ^, the 

 inferior limit of the integration of — cos y cos [(y — ft) -|- y] dy 

 remains constant at ^ tt — /?; and the superior limit of the other inte- 

 gration becomes ^ tt — (v — 2/3). The result of each integration 



is now 



i [sin ft cos (v — 2 ft) — ft cos {v — /3)]. 



