Now let I, be the luminosity of s, and I, that of the surface RT as seen in the medium by the light coming originally from s.

R

N

FIG. 232.

I1 is the luminosity RT must have to send back to s the same amount of light as s sends to it—that is, as R T sends on into μ1⁄2. If s1 is the area of the surface R T, the quantities of light received and sent on by it are—

I111 cos i, and I ̧ ̧1⁄2 cos î

And these are equal. Thus from above

I1 : I2 = μ‚2 : μ3.

In the same manner, the luminosity of the second surface across which the rays pass, as seen in the third medium of index μ, is proportional to μ2; and so on.

If the rays on passing across the last surface form a true image of s, so that the combination of surfaces is aplanatic for s, the brightness of this image is the same as that of the last surface; for it subtends the same solid angle at the point of view as the portion of the surface which sends the same quantity of light to the point of view. Thus, if μ, ' are the refractive indices of the medium in which the object is and that in which the eye is which sees the image, and I, I'ˇthe luminosities of the object and image—

1 : I' = μ2 : μ22.

ANSWERS TO EXAMPLES

I.

2 feet; 3 feet.

I.

2. 50 cms.

4. 27'5 nearly.

5. Inclined to the line joining it to the point at an angle whose sine is

π