218 CELESTIAL MECHANICS. I.V. 2l. 



the squares of the areas of the three trian- 

 gular faces of the soHd angle. 



The area of the face between a and & is i a6, and the 

 perpendicular falling on its base from the solid angle is 



' —-: but this perpendicular must be perpendicular 



to the third side c, and the square of the hypotenuse of the 



triangle lyinff between them must be c^H --, which 



° aa + ob 



multiplied by the square of the side to which it is perpen- 

 dicular, or a^-\-lr, must be the square of twice the area; 



consequently the square of the area is i < {a^ + h^) c^ + a^h^ > 



rzia^S^ + AaV + i&V, which is the sum of the squares of 

 the areas of the three faces. 



328. Lemma D. The sum of the squares 

 of the projections of any area, on three ortho- 

 gonal planes, is equal to the square of the 

 area itself. 



For the projection of the area on each plane is to the 

 original in the same proportion, as the whole face of the 

 parallelepiped is to the whole oblique section: the pro- 

 portion of the areas being determined by the inclination of 

 the planes, whatever the form of the area projected 

 may be. 



329. Lemma E. The cosine of the incli- 

 nation of the section to either of the faces 

 will be expressed by the area of that face di- 

 vided by the area of the section. 



