2S 



arrived in my own researches, before I was aware of the existence of 

 his. To investigate the connexion of these curves and surfaces with 

 the characteristic functions V and W, let us consider the conditions 

 which must be satisfied, in order that a curve having for coordinates 

 x", y", z', should be touched by an infinite number of rays of the sys- 

 tem. Let x, y, z, be the coordinates of any point on such a ray, and 

 g its distance from the point of contact a/' y" 2'', in such a manner 

 that we may put 



x = x" + «5, y = y"+^{, « = z" + y5, 



and therefore 



we shall then have 



3x' = Jx — a, [ciix -\- fihy + yiz) ■=. {3« , *\ 



V = 3y — /3 («Jx + /SSy + y3«) = jS/S , K (A') 



iz' = lz — y («Sx -{- fiiy -f- yiz) — ^"iy , ) 



assigning to h' Sy' h' the same meanings as in the fifth number, 

 and observing that by the nature of of' y" z", the variations iaf' dy" h" 

 are proportional to a, 0, 7, so that 



3x" = « («5x" + fiiy'i + ySz"), 

 iy" = /J («3x'' + fiiy" + yiz") , 

 h" = y {^x" + ^y" -f yiz") . 



The formula (A') give 



3y 3y 





y £■ having the same meaning as in the fifth number : and these 



