386 Mb airy, on the INTENSITY OF LIGHT 



which, on ascertaining the deviation (from a fixed direction) of the rays 

 reflected or refracted, we find, on proceeding in the same direction 

 along the reflecting or refracting surface, that the deviation increases 

 to a certain amount and then diminislies, or vice versa. In this case 

 tlie caustic consists of t'vo imconnected infinite branches in opposite 

 directions, with a common asymptote parallel to the position of maxi- 

 nuim or minimiun deviation of the rays. To investigate this case, we 

 sliall examine the form of the front of the wave immediately after 

 leaving the reflecting or refracting surface, and shall measure the 

 lengths of paths of light from that front. In fig. 3, let A be the 

 point at which the asymptote intersects the front of the wave (which 

 will be the same as the point of the front where the deviation is 

 maxinuim or minimunij whose coordinates are and b : let X be any 

 other point in the front, whose co-ordinates are x and y, (x being 

 measured from the asymptote and tj parallel to it:) and /; and q the 

 co-ordinates, similarly measured, of any point P near the asymptote. 

 If the length AX be called *, and the angle made by the tangent at 

 X with the tangent at A be called 0, then the condition that the 

 deviation of the direction of the rays from a fixed direction (or jthe 

 deviation of the tangent to the front of the wave from another fixed 



direction,) is maximum or minimum gives = -, « being some con- 

 stant. Observing that -j- = cos 6, and -^ = sin 6, we get with suf- 



ficient approximation x = .v, ij = h + = & + ^— ^: and the front of 



the wave is therefore a cubical parabola. The distance of the point F 

 from JT 



=. V{x -pT + {y- qf = \/f - ^px+x' + (i - qy + \j^ ^. 

 and, expanding this to the third power of x, and putting c" for p' + (b -qf. 



