280 [Nov. 27, 



factor, or we may write HU=U . PU. The function PU, which for 

 the developable replaces, as it were, the hessian HU, is termed the 

 prohessian ; and since, if r be the order of U, the order of HU is 

 4r 8, we have 3r 8 for the order of the prohessian. If r=4, the 

 order of the prohessian is also 4 ; and in fact, as is known, the pro- 

 hessian is in this case =U. The prohessian is considered, but not 

 in much detail, in Dr. Salmon's ' Geometry of Three Dimensions ' 

 (1862), pp. 338 and 426 : the theorem given in the latter place is 

 almost all that is known on the subject. I call to mind that the 

 tangent plane along a generating line of the developable meets the 

 developable in this line taken two times, and in a curve of the order 

 r 2 ; the line touches the curve at the point of contact, or say the 

 ineunt, on the edge of regression, and besides meets it in r 4 points. 

 The ineunt, taken three times, and the r 4 points form a linear 

 system of the order r 1, and the hessian of this system (considered 

 as a curve of one dimension, or a binary quantic) is a linear system 

 of 2r 6 points ; viz. it is composed of the ineunt taken four times, and 

 of 2r 10 other points. This being so, the theorem is, that the 

 generating line meets the prohessian in the ineunt taken six times, in 

 the r -4 points, and in the 2r -10 points 



it is assumed that r=5 at least. 



The developables which first present themselves are those which 

 are the envelopes of a plane 



(,*, ...-$, i)"=o, 



where t is an arbitrary parameter, and the coefficients (a, b, . . .) are 

 linear functions of the coordinates ; the equation of the developable is 



disct(, ft..X*, l) n =0, 



the discriminant being taken in regard to the parameter t. Such 

 developable is in general of the order 2n 2 ; but if the second co- 

 efficient b is =0, or, more generally, if it is a mere numerical 

 multiple of a, then a will divide out from the equation, and we have 

 a developable of the order 2w 3 : the like property, of course, exists 

 in regard to the last but one, and the last, of the coefficients of the 

 function. "We thus obtain developables of the orders 4, 5, and 6 

 sufficiently simple to allow of the actual calculation of their probes- 

 sians. And the chief object of the present memoir is to exhibit these 

 prohessians ; but the memoir contains some other researches in rela- 

 tion to the developables in question. 



