468 The Rev. Geoege Salmon 011 the Degree of the 



It has a cuspidal line of the degree m, and an ordinary double line of the 

 degree a;. The simple line of contact (a) consists ofw right lines. Each of 

 those right lines meets the edge of regression once, and the line x in r - 4 

 points (see "Cambridge and Dublin Mathematical Journal," vol. v. p. 25). 

 The lines m and x intersect at the a points, which are the points of contact of 

 stationary planes of the system ; for, since there three consecutive lines of the 

 system lie in the same plane, the intersection of the first and third of these 

 must belong to the line .r, which is the locus of the points on two consecutive 

 lines of the system.* The tangent cone to the developable breaks up into n 



« f « 1 ^ 



planes ; it has, therefore, no cuspidal lines, and — ^ij — ^ double lines. 



We have then the following table. The letters on the left-hand side of the 

 equation refer to the notation of the preceding theory ; the letters on the right- 

 hand to the notation in the papers on developable surfaces just referred to: 



n = r, a = n, b = x, c = ot, /) = n(;'-4), a = n, k = 0, p = j3, h = h, i = a, 



and the quantities t, 7, k remain to be determined by the present theory. On 

 substituting these values in the equation (A) and (D), we obtain the following 

 system of equations : — 



ti(r-2)=n\2 + (r-4:)], 



X (?--2) = n(?--4) + 2/3-t-37 + 3i, 



m(r-2) = 2?z-f 4/3-f 7, (E) 



n{r-'2) (r-3) = 7t|(n- l)-t-2.T-l- 3m - 4 (r- 4) - 9|, 



X (r-2) (r-3) = 4/;-^Mar-f3ma--9j3-67-3a-2?2(7--4), 



m (r - 2) (r - 3) = 6^ H- mn + 2mx - 6/3 - 47 - 2a - 3n. 



The first of these equations is verified immediately, and the fourth by the help 

 of the equation given by Mr. Cayley (" Cambridge and Dublin Mathematical 

 Journal," vol. v. p. 21), 



2x + 3m + n = r(r- 1). 



We may determine 7 either from the third or from the sixth equation. That 



* It was the consideration of this case which led me to include in the preceding theory the 

 points I, of wliich I have never happened to meet with any other instance. 



