382 Mr airy, ON THE INTENSITY OF LIGHT 



5. Now the definition of a caustic in Geometrical Optics, is " the 

 locus of the ultimate intersections of reflected rays : " and therefore, for 



every point of a caustic, ~~- = and -j -j-~ = when that value of 



X is used which corresponds to the point of the reflecting surface, from 

 which the light is reflected to each particular point of the caustic. But 



,' , is not necessarily = 0: and in general its value is finite. For 



if, in fig. 2, Ave take a point P' of the caustic nearer to the reflecting 

 surface than P, and if X' is the corresponding point of the reflecting 

 surface ; then we know from the geometrical theory of caustics, that 



SX' + XP + PP = SX + XP. 



Now if we join X' with P, it will be evident that 



XP < XP + PP. 

 Therefore, SX' + XP < SX + XP, 



or r' < V. 



Similarly, if we take a point P" on the caustic further from the 

 reflecting surface than P, and X" for the corresponding point on the 

 reflecting surface, 



SX" + XP" = 5X + XP + pp\ 

 But X P + PP' > XP. 



Therefore SX" + X P + PP > SX + XP + PP. 

 Or SX' +X'P>SX+XP, 



or F" > F; 



consequently the first differential coefficient of F which has a finite 

 value is of an odd order : and as, in the general case, we must, from 

 the very meaning of the word general, take those conditions which 

 require the smallest number of peculiar equations, we must fix on the 



