

Dr. W. E. Suinpner on the Diffusion of Light. 87 



bench and not to RP 2 , we must multiply this expression by 

 cos 02 to get the effective illumination due to the element 

 dA. Finally we have for the total illumination the integral : — 



1 = — -^ — ol cos 3 (f>i cos 4 <f> 2 dA. .... (9) 



in which the angles <£ 1? (j> 2 are related by the equation 

 # x tan (pi = x 2 tan (£ 2 . 



When the area A is circular, with as its centre, this 

 integral reduces to 



Irr^fcos'fcd.sin 2 ^. . . . (10) 



The value of this integral can be readily evaluated, but it does 

 not lead to a convenient formula, and as it was found prac- 

 tically preferable to fix the lamp to the same slider as the 

 photometer, and at the same distance from the screen OR, 

 we may put 



4>l = <£ 2 = <£> 



x x = # 2 = «r, 

 and (10) then reduces to 



i=^ |[i -«*•«, (ii) 



in which <£> is the semiangle of the cone with base A and 

 height x. 



This expression is rendered more convenient for purposes 

 of calculation by taking advantage of the fact that A/ttx 2 , or 

 tan 2 <£, is a small quantity. By neglecting tan 6 </> compared 

 with unity we obtain 



g[l-COS 5 <£] =^> 



where 



X=~\l'75 + '4,S—., ) (12) 



and in most cases it will be found that the third term in this 

 expression is negligibly small compared with the sum of the 

 other two. 



The value of I found in (11) may be equated to k/y 2 when 

 the photometer is in the position of balance, and on doing so, 

 we find for rj the value 



»-i^ (13) 



which reduces to (8) when x is large. 



