Prof. Stokes on Internal Radiation. 479 



and applying this formula to the sphere by replacing s V l+p 2 +q 2 

 by 1, v by 0, and p by 1, we have 



cos i^cp^JTd^drj, 



the double integral being taken over the projection of the corre- 

 sponding small area of the sphere. 



Now by the well-kDOwn formula for the transformation of multiple 

 integrals we have 



and therefore 



cosily _ scosv Vl+j3 2 +g 2 



cos \cty~~ 3 /^£ ^H_^1L ^!l\ 

 ^ \dx dy dy dx) 



But the first of equations (4) gives 



jy (px+qy—z)dp—p(xdp+ydq) 

 (px + qy—zf 



_ {(W— z ) r -pyA dx+ {(qy-z)s—pyt) dy 



(px + qy—zf 

 Similarly, 



\(px—z)t—qxs\dy+^(px—z)s—qxr^dx m 



(px + qy—zf 

 Hence 



dx dy dy dx (px + qy — zf 

 where 



V = \{iy— z ) r —py s } {(px—z)t—qxs\ — {(qy—z)s—pyt\ {(px— z)s— qxr\ 

 = {{qy— z)(px— z)—pqxy\(rt— s 2 ) 

 = z(z — px — qy )(r t — s 2 ) . 



Hence 



cos zfy __ z cos v V 1 4-y + g2 ^ 2 _ px _ qy y ^ 



cosi 1 %<p 1 p 3 z(rt—sr) 



But if to- be the perpendicular let fall from O on the tangent plane 



atP, 



z—px—qy —s/\ +p 2 + q 2 % m i 

 and therefore 



cos ity _COSv.tr 1 (l+p 2 + q 2 f 



cos^^ p 3 rt—sr 



But z<r=p cos v. Also the quadratic determining the principal radii 

 of curvature at P is 



(rt-fy + (&c.)v + (l +p 2 +qj=0 ; 



