m 



which shews that there are in general an infinite number of osculating 

 focal systems corresponding to any given ray, that is, an infinite 

 number of different values for the arbitrary parameter which enters 

 into the expressions of 



'*'" «- yiv' yw yiv ' "^'^ ' 



according to the infinite variety of values that we may assign to the 

 ratio 



niviid ,,:: ,,\t ,.x bus ,i*i)'jo't _ > -'di ■_ 



but that the values of this arbitrary parameter, which do not change 



for an infinitely small alteration in the ratio on which they depend, 



are determined by the following equations : 



which give, by elimination, 



The sj^stems that correspond to these extreme values of the arbitrary 

 parameter, we shall call the extreme osculating focal systems ; and 

 since, by the nature of the functions W, W, the equations {W') are 

 equivalent to the formulae {R'), the foci of these extreme osculating 

 systems are contained upon the caustic surfaces : and the ratios of 

 3a, 5/3, ly, in these extreme systems, are the same as in the develop- 

 able pencils. 



12, Let us now consider the law of the variation of the focus of 

 the osculating system, between its limiting positions. This law is 

 analytically expressed by the formula ( U') ; in which we may geo- 



(W) 



