470 The Rev. George Salmon on the Degree of the 



that the given line should meet the curve U V again. This condition will be of 

 the degree {k— 2) {I — 2) in wyzw, and of the degree Ik — i in x'y'z'io' . But 

 since this condition must be satisfied for every point of the given line, if we 

 eliminate a;'?/' j'z^' between it, the two equations of the right line, A£/'=0, 

 AF= 0, and the equation of an arbitrary plane, ox + /3y + 7Z + 5ii) = 0, we 

 shall have a result which will be of the form 



n {ax + /3y + 7^ + 8u;)'*^. 

 n will then be of the degree 



{k - 2) (Z - 2) + (Ik _ 4) (A + Z - 3), 

 and the intersections of the surface represented by n = with TJ., V will give 

 the points 7. If then we write kl = q, k + l =p, the number of points 7 will be 



q (pq -2q-Qp + 16). 



But I have shown (" Cambridge and Dublin Mathematical Journal," vol. v. 

 p. 32) that in the same case 



m = q, n = 3q(p-3), r = q (p - 2), /3 = 0, 



by the help of which values the equation 



m (r - 2) = 2n + 4/3 + 7 



is satisfied identically. The case of developable surfaces, thus examined, 

 proves, I think, that the numerical coefficients in at least the first and third of 

 equations (A) and (D) have been rightly determined. It is scarcely necessary 

 to observe, that the singularities here noticed may be sometimes replaced by 

 others of a higher order. For example, the developable, whose edge of re- 

 gression is the line of intersection of two sui'faces of the second degree, has no 

 " point on three lines," but has, instead, four " points on four lines." 



As a further illustration of the theory of developable surfaces, I take the case 

 of the developable which is the envelope of the variable plane 



At'^+ fxBt'^-' + ''-^^^ Ct^-' + &c. = 0, 



where t is a variable parameter. 



This surface has been elsewhere discussed (" Cambridge and Dublin Ma- 

 thematical Journal," vol. iii. p. 169 ; vol. v. pp. 46, 152). Its characteristics I 

 have there stated to be — 



i 



