478 Royal Society : — 



to the normal to II. Then shall the ratio of cos id(j> to cos i'ty be of 

 the form [P] : [P'], where P depends only on the first surface and the 

 position of P, and [P] only on the second surface and the position 

 of P'. Moreover, if either surface be a sphere having its centre at O, 

 the corresponding quantity [P] or [P'] shall be constant. 



It may be remarked that the two surfaces may be merely two sheets 

 of the same surface, or even two different parts of the same sheet. 



Instead of comparing the surfaces directly with each other, it will 

 be sufficient to compare them both with the same third surface ; for 

 it is evident that if the points P, P' correspond to the same point P x , 

 on the third surface, they will also correspond to each other. For 

 the third surface it will be convenient to take a sphere described 

 round O as centre with an arbitrary radius, which we may take for 

 the unit of length. The letters P x , i v fa will be used with reference 

 to the sphere. 



Let the surface and sphere be referred to rectangular coordinates, 

 O being the origin, and II the plane of xy. Let x, y, z be the coor- 

 dinates of P ; ^, 77, 4T those of P x . Then x, y, z will be connected by 

 the equation of the surface, and £, 77, £ by the equation 



According to the usual notation, let 



dz __ dz __ d?z _ cPz d?z _ 



dx~^' dy~^ y dx 2 ~ ' dxdy~" ' dy 2 ~ 



The equations of the tangent planes at P, P x , X, Y, Z being the 

 current coordinates, are 



Z-z=p(X-x) + a(Y-y) s 



£X+>?Y-KZ=1;: 

 and those of their traces on the plane of xy are 



pX + qY=px + qy—z, 



SX+>?Y=1; 

 and in order that these may represent the same line, we must have 



l=—r^~> ,-- -^ . . . (4) 



px-\-qy—z px + qy—z 



To the element dxdy of the projection on the plane of xy 

 of a superficial element at P, belongs the superficial element 

 dS= s/\-\-p 2 +q 2 dxdy, and to this again belongs the elementary 



solid angle § — , where p=OP, and v is the angle between the 



normal at P and the radius vector. Hence the total solid angle 



COS V 4 i 



within a small contour is — ~\/l+i ?2 +<Z 2 l \dxdy, the double in- 

 tegral being taken within the projection of that small contour. Also 

 cos i=- • Hence 



.. #cosv s (T\ , 



cosid0=— 3 — l+p 2 -)-q 2 \\dxdy ; 



