86 Mr. R. Potter on a particular Modification 



To find the simultaneous positions of the luminiferous sur- 

 faces on the axes of the pencils after the two refractions, which 

 are supposed to depart simultaneously from a and b. 



We have first, ec or cf=cr or cs x sine incidence, 

 and r d or s d = c r or c s x sine refraction 



^ = r ^ X lmT = rdX[x 

 and ec = 2a tan ii hence rs = 2 xrrf = 4a 



Now let the velocity of light in air be to the velocity in glass 

 as 1 to ;«. Then when the upper ray arrives at s, the lower 

 one will be at a point in its path, with respect to the point^ 

 represented by this expression : 



/ s\ rs 



(e c + cj) co 



m 



tan * 



or, 4 a tan i oc 4a = Aa tan 



i (1 CV3 j 



\ my. ) 



On the undulatory theory m is supposed to be the reciprocal 



of the refractive index; or we have m = — . Then the 



above expression, which we may call the difference of the paths, 

 or the difference in the simultaneous positions of the lumini- 

 ferous surfaces counted on the axes of the pencils, becomes this: 



A paths = 4a tan i ( 1 — — J == 4a tan i (o) 0. 



On the Newtonian hypothesis, that the velocity is directly as 

 the refractive index, we have m = ju,, 



and A paths = 4a tan i M j = 4a tan / ( 1 A 



= 4a tan i \—~^ — )• 



The last preliminary question to be examined, or that of 

 determining the curve of the principal section of the lumini- 

 ferous surface after refraction, requires the introduction of 

 differentials and the method of polar coordinates. Taking 

 two rays ao ap (fig. 3), as indefinitely near each other, and 

 diverging from the point a, we may take the indefinitely small 

 and perpendicular distances between the rays in the prism pn 

 and rs as equal to h and h! ; and now calling the variable 

 angle of incidence on the first surface <p, the corresponding 

 angle of refraction <p' ; the angle of incidence on the second 

 surface \|/, and the corresponding angle of emergence 4> • — 



