92 



vrefiball thus phtain the relation between («, /S) vfhich characterises the rays that pass through 

 the given curve : and substituting, in this relation, the values of («, fi) considered as functions of 

 {x, y, z) we shall have the equation of the pencil. 



In tliis manner we can determine the sliadow of any opaque body, produced by the rays of a 

 given reflected system, if we know the equation of the body, and that of the skreen upon which 

 the shadow is thrown ; we can also determine the boundary of light and darkness upon the body, 

 which is the curve of contact with the enveloping pencil ; and if we consider visual instead of 

 luminous rays, we can determine, on similar principles, the perspective of reflected light, that 

 is, the apparent form and magnitude of a body seen by any combination of mirrors ; at least so 

 far as that form and magnitude depend on the shape and size of the visual cone. 



£25. "1 Besides the general analytic expression 



which represents all the pencils of the system, by means of the arbitrary function (f), we can 

 also find another analytic expression for those pencils, by eliminating that arbitrary function, 

 and introducing instead of it the partial differential coefficients of the pencil of the first order. In 

 this manner we find, by differentiating (A'') for {x) and (_y) successively, and eliminating the dif. 

 ferential coeiBcient of the arbitrary function (y), 



£21 ££ ^(£L.X = S '^"'^ '^^ _ — ^'^ ? ^ 



dx^ ' drf- \dx.dy) (_ dx.dy ' dy.dz dy^ ' dx.dz 5 dx 



<i £v_ £_y__(py_ _££.? ^. 



\ dx.dy ' dx.dz dx^ ' dy.dz $ ' dy ' 

 and since the general relation a'' -|- /3^ + y' = 1, that is 



©'-©■+©' = -• 



gives by differentiation 



0. 



d^v d^v d^y 



J. a, L «, — 0, 



dx^ ^ ^ dx.dy ^ ^' dx.dz ' 



d'V d^V d^' 



and therefore 



' dx.dy dt^ dy.dz 



d'V d'V dW dW _»_ (d^V dW ^/ dW yi 



dx.dy ' dy dz dy^ ' dx.dz y' \dx^ ' dy^ \dxdy J 5 



dW d^V d^V d^V _ /3 C£F d'V / dW y ^ 



1y.dz ~ y' Idx^' dy^ \dx.dy) \ ' 



dx.dy dx.dz dx'' dy. 

 the partial differential equation of the pencils becomes finally 



