466 The Rev. George Salmon on the Degree of the 



[aU] — ab — lp 



[ac] = ac — 3(7 (C) 



{he ] :::. ic - 3/3 - 27 - l* 



Substituting this value for [6c], we have 



2c = (a-26-3c) (n-2) (?i-3) + 8A + 18A + 126c-36/3-247-12J, 



and since, if the curve had no multiple lines, twice the number of double edges 

 in the tangent cone would be (a + 26+ 3c) («-2) (w-3) ; the diminution in 

 28, caused by the double lines, is 



(4/j + 6c) (w-2) (?i-3)-8i-18A-126c + 36^ + 247+12i. 



By the help of the equations (C), the equations (B) may be written in the fol- 

 lowing form, which is sometimes more convenient: 



a{n-2) (n-3) = 28 + 2a6 + 3ac-4^-9<7, 



6(n-2) (n-3) = 4^+ a6 + 36c-9/3-67-3i-2p, (D) 



c(k-2) (n-3) = 6A+ ac + 26c - 6/3 - 47 - 2z - 3<7. 



It is easy now to find the efiFect of the lines 6 and c on the degree of the reci- 

 procal surface. If the degree of a cone diminish from m to m — I, that of its 

 reciprocal will diminish from rn {m — 1) to (m -I) (m — Z— 1) ; that is to say, 

 will diminish hy l{2rn — l-l). In the present case m = n^ — n, and Z = 26 + 3c. 

 The diminution then in the degree of the reciprocal, arising from the diminu- 

 tion in the degree of the tangent cone, is 



(26 + 3c) (2n2-2w-26-3c-l). 



We must subtract from this three times the diminution in the number of cusps, 

 together with twice the diminution in the number of double edges, and we find, 

 for the total diminution of the degree of the reciprocal surface, 



* It is proper to observe, that if the surface have a nodal curve (6), but no cuspidal curve, 

 there will still be a determinate number i of cuspidal points on the nodal curve, and the equation 

 given will receive the modification [a6] = ab-2p- i. As, however, the quantity [nJ] is eliminated 

 from the equations in finding the diminution in the degree of the reciprocal surface, the ultimate 

 result is not affected. 



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