300 Messrs. EL E. Ives and E. F. Kingsbury on the 



foe exactly complementary ; in other words, their range of 

 fluctuation of value must be equal. But while the actual 

 range is alike, the fractional range, in terms of the mean 

 value, which on our theory determines the disappearance of 

 flicker, will be small for the large white opening, and large 

 for the small white opening. It is easy to see that (the 

 difrusivity being fixed) the critical speed must increase pro- 

 gressively from zero to 360 degrees white opening to large 

 values for small openings. But, according to our theory, 

 the difrusivity varies with the illumination and in the same 

 direction. It is to be expected that it is the average illu- 

 mination (or, strictly speaking, the average brightness), the 

 product of illumination and white opening, that determines 

 the dirTusivity. Accordingly we may expect the diffusivity 

 to decrease from a maximum at 360 degrees to very low 

 values for small angles. Lowered dirTusivity calls for lower 

 speed of alternation to produce the same range. We therefore 

 have, according to our theory, two opposing effects : one, 

 which w T e may call a contour effect, calling for increasing- 

 speed ;is the white angle decreases ; the other, the dirTusivity 

 effect, which calls for decreasing speed as the white opening- 

 becomes small. If our theory is correct, the resultant of 

 of the two should have a maximum at 180°. The adequacy 

 of these effects to explain the experimental facts can, of 

 course, be determined only by quantitative calculations, which 

 we shall now attempt. 



Here we cannot, as in the case formerly treated, assume 

 a simple sine curve stimulus. We have therefore approached 

 the problem in a different manner. We have first developed 

 the complete Fourier series expansion for unequal light and 

 dark intervals, with abrupt transitions, as follows: 



21 r i 



I s = I<f>-\- — sin 7T(/>cos cot-\- -sin 2tt(/>cos 2cot 



IT L ^ 



+ -sin o7T(j) cos Scot + .... . (8) 



We have then combined this, in the same manner as in 

 our earlier paper, with the fundamental Fourier conduction 

 equation. 



The result is the following infinite series: 



9T r -XV -^ / / \ 



I ? =L/> + — e -'K s i n7r 0cos( cot — X\/ -™\ 



+ - } e~ E sin2 7 r(/)COs/2w^-XA v /^)+ ....], (9) 



