WHICH REFLECT LIGHT. 321 



through the angle 2(i— i'). The corresponding intensity on the 

 firmament, therefore, no longer belongs to the element M, but 

 is distributed over an infinitely small ring PHQ, round M, with 

 the radius MH = 2(i— i'). Now if the distance of the point M 

 from CD is smaller than 2{i—i'), one portion of the ring lies at 

 the other side of D ; and when we conceive that through the 

 gradual enlargement of the ring the latter is pushed over CD, 

 we see at the same time that the bit JK, which lies between 

 the lines of the angle EMF=y8, must have passed through EF. 

 The luminous intensity of this bit is to that distributed over the 

 entire ring as y3 : 27r ; and the magnitude of the angle jS, which 

 we may regard as very small, is determined by the angle a which 

 the straight line MG encloses with the normal NG, and the 

 length MG=^; that is, 



r, dy . cos u 



and thus we obtain the luminous intensity comprised in the bit 

 JK, 



g.lTT ^ ' L 2sm2(^ + ^0 2 tan2(i + i') J ^ ^ 



It will be of advantage to combine the consideration of the dw 

 by M wdth that of another situated at an equal distance from CD 

 and exactly opposite, say by the point M' with the coordinates 

 od and y. The action of this element is opposed to that of the 

 foregoing one, inasmuch as a portion of its light is sent towards 

 the left through EF, and the difference of both actions may 

 therefore be regarded as the total action of both. Treating this 

 element exactly as the former, we obtain the same formula as 

 (14), with the exception that oc* stands in the place of x. Finding 

 the difference of the two formulae, and remembering that 



or'— a; = 2^ cos a, 

 we obtain 



dij.Q,m^u_, ^. . .^.r, Isin^fi-i') Itan^i-i')!^ ,,^, 

 ~ dco, 2 smi cos tdt\ l—7:-r-gyT-—.,i—-^ — o)' , ■, { • (15) 



The same expression is found for every pair of elements which 

 stand symmetrically on both sides of CD, and whose distances 

 from G are not greater than 2{i—iS), We must therefore inte- 

 grate it for the semicircle described round G and over CD. For 



