82 Dr. W. E. Sumpner on the Diffusion of Light. 



area in the direction normal to the surface. The illumination 

 produced at a point by a surface of brightness B, subtending 

 a (small) solid angle O at the point, is BO, and if a surface, 

 containing this point, have its normal inclined at an angle (f> 

 to the axis of O, the illumination of this second surface due 

 to the first is Bfl cos </>. 



If B is the brightness of a surface rendered luminous by 

 reflexion, if rj is its (diffused) reflecting-power, and if I is 

 the illumination of the surface, then 



ttB = v I (1) 



This relation follows from the assumption of the law of 

 cosines, viz. that the candle-power (per unit area) of a bright 

 surface is B cos <j) in a direction making an angle <f> with the 

 normal. The right-hand member of (1) is by definition 

 equal to the whole light reflected from unit area of the sur- 

 face, and must be equal to the integral of B cos </> dfl for all 

 directions on one side of the surface. The value of the 

 integral is easily seen to be 7rB. 



The brightness of a body, as just defined, is directly pro- 

 portional to the illumination of the image of the body on the 

 retina of the eye, and the word may thus be quite justly used 

 in the ordinary physiological sense of the term. Similarly 

 the law of cosines, just alluded to, is merely another way of 

 expressing the fact that the sensation of the brightness of a 

 diffusing surface is the same from whatever direction this 

 surface may be viewed. For the amount of light received 

 by the eye, and concentrated on the image on the retina, is 

 simply the product of the area of the pupil of the eye and 

 the illumination at the surface of the eye due to the bright 

 object. The former factor is constant for different directions 

 of view, and the latter must vary as the solid angle subtended 

 by the object at the eye, since the area of the image on the 

 retina is a measure of this solid angle. The illumination at 

 a point at which a surface of brightness B subtends a solid 

 angle fl is not necessarily BH for all inclinations of* 12 to the 

 surface, unless the law of cosines is fulfilled ; and, if this law 

 is fulfilled, it follows that the illumination of the image of 

 this surface on the retina of the eye is constant at all dis- 

 tances and inclinations. The eye is a good judge, and is 

 indeed the only judge, of quick variations in brightness, and 

 the cosine law is always applicable to diffusing surfaces, the 

 appearance of which does not alter as the eye moves past 

 them. Any divergence from this law is negligible so far as 

 its influence on illumination is concerned. 



If Q is the quantity of light radiated per second by the 



