34 



vanish when « = 0, /3 z= 0. If therefore v'' be positive, and if we 

 denote by H^ the greater of the two values Rj , R^, that is the one 

 nearer to positive infinity, we shall have by {F"), for all other values 

 of R, 



K>R,, R<K, , («">0); (G") 



SO that in this case the foci of the osculating systems are all ranged 

 upon that finite portion of the ray which lies between the caustic sur- 

 faces. If, on the contrary, tf' is negative, then the two differences 

 R — Ri and R — R^ are both positive or both negative, so that 



1^1^ >0, («" <0); (H") 



in this case, therefore, the foci of the osculating systems are all con- 

 tained upon the remainder of the ray, that is upon the two indefinite 

 portions which lie outside the former interval. And in each case, 

 the distances of the focus of any osculating system from the two 

 points in which the ray touches the two caustic surfaces, are propor- 

 tional to the squares of the sines of the angles which the plane of 

 osculation makes with the two tangent planes to the developable pen- 

 cils. In the foregoing investigations we have supposed that TV, and 

 its analogous function W, which we consider for symmetry as homo- 

 geneous, are put under the form of functions of the dimension zero ; 

 a supposition which permits us to adopt the expressions {K) for the 

 partial differentials 



iw iw iJV 



i» a/3 3y 



instead of the less simple and more general expressions given in the 

 fourth number: but if we had assigned any other value to the dimen- 

 sion i, in those more general expressions, we should have deduced the 

 same results respecting the law of osculation. 



