On FresneVs Convection Coefficient. 149 



■co-ordinates, and let Ox in the plane of the paper, Oy per- 

 pendicular to the plane of the paper, and Oz be the rect- 

 angular axes. Thus the normal to the refracting surface 

 is 0.:. Draw any line through representing the incident 

 ray whose direction cosines are Z 1? m 1? w l5 and let it meet the 

 sphere whose centre is C 2 in P 1# In like manner draw 

 another line representing the refracted ray whose direction 

 cosines are l 2) m 2 , n 2 , and meeting the other sphere in P 2 . 

 The points Pi and P 2 are, of course, not generally in the 

 plane of the paper. If, now, tangent planes be drawn to 

 the spheres at P x and P 2 , they must intersect in a line lying- 

 in the plane of xy. 



We now assume that v 1 = k 1 v and v 2 = K2V, and in the work 

 squares of t\ and v 2 will be neglected. Let the angle 

 .xOv = (j). 



We have OPi = C : — r^ cos ^ + n x sin <£), 



and the co-ordinates u l9 f-} l} <y x of P : are 



u i = c i l 1 — Vil x (liCOs<p + iii^incf)). 

 /3 2 = ciThi — ?v n iGi cos (j> + n x sin <f>) , 

 ry x = c 1 n 1 — v i n l (/ : cos cj) + ?i 1 sin <£). 

 In like manner the co-ordinates a 2 , j3 2 . <y 2 of P2 are 

 u 2 = cj 2 — v 2 l 2 (l 2 cos <fi + n 2 sin <j>) , 

 ^ 2 = c 2 m 2 — v 2 m 2 (Z 2 cos cf> + n 2 sin ^>), 



On writing down the equation of the tangent plane at P t 

 we find that it meets the plane of xy in the line 



/ il + ^coscftx ^ =1 



and that the tangent plane at P 2 meets the plane of xy in 

 the line 



/ 1 2 Vo cos d>\ 



^ = 1 



We thus have 







m l c l {^2 



m 2 c 2 fii 



p> 



where fi l is the absolute index of refraction of the first medium 

 and fjL 2 that of the second, fi being the index of refraction 

 from the first medium to the second. 



