IN THE NEIGHBOURHOOD OF A CAUSTIC. 383 



first coefficient of an odd order which has not yet been fettered by any 

 equation ; and therefore in the general case, for" a point in a caustic, 



J ., has a finite value. 



It may possibly happen at singular points that ^^ and ^^^ vanish 

 , ,^ dar die' ' 



dl'i V) 

 and that -^— has a finite value : but of these peculiar cases T intend 



to take no further notice. 



6. By pursuing this train of investigation we should find that at 



a finite value. I shall not however pursue this subject further. 



7. The conditions then which hold, with reference to any point 

 of a caustic in general, are these : If T be measured from the origin 

 of light to any point of the reflecting surface and then to the given 

 point of tiie caustic : in the case of the point of the reflecting surface 

 coinciding with the corresponding point of reflexion, 



dx 



d\V) 

 daf 



d\r) 



due 



= 0, 



= c. 



C being a finite function of x, y, p, and q. The sign of C may be 

 thus found. In the case assumed in (5) and represented in fig. 2, 

 V was < F and V" > V\ if then V implies that x is diminished, or if 

 X is measured from the convexity of the caustic, C is positive. If x is 

 measured towards the convexity of the caustic, in that case C is negative. 



8. The value of C may thus be found. Draw T^Q' perpendicular 

 to PX': then Q'X' may be considered equal to V X' (it will differ from 

 it only by quantities depending on the fourth power of PP' or XX") ; 



