543 



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



. (4) 



z px+qyz 



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

 of a superficial element at P, belongs the superficial element 

 , 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 





within a small contour is xlH^ 2 +2 2 lda%, the double in- 

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



cos t=- Hence 



Z COS V 



cos * = 



and applying this formula to the sphere by replacing z 

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



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

 sponding small area of the sphere. 



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

 integrals we have 



and therefore ___ 



cos i^ _z cos v V 1 +/ + 



s/^L ^?_^. ^\ 



' \dx dy dy dx) 

 But the first of equations (4) gives 



yz)dpp(xdp+ydq) 



Similarly, 



\(px z)t qxs] dy+ \(pxz)sqxr\ dx 



