1871.] 



Theory of Reciprocal Surfaces. 



23 



2. In the formulae 



q=b-~b-2k~3y-6t, 

 r= c 2 —c-2h-3fi 3 



it is assumed that the nodal curve has no actual multiple points other than 

 the t triple points, and no stationary points other than the y points which 

 lie on the cuspidal curve ; and similarly that the cuspidal curve has no 

 actual multiple points, and no stationary points other than the /3 points 

 which lie on the nodal curve ; and this being so, q is the class of the nodal 

 curve and r that of the cuspidal curve. But we may take the formulae as 

 universally true ; viz . q may be considered as standing for b 2 — b — 2 k — 3y — 6 1, 

 and r as standing for c 2 — c— 2h~3(3; only then q and r are not in all 

 cases the classes of the two curves respectively. 



3. In the formulae No. 6 et seq. s introducing the new singularity w, we 

 have as follows : — 



(a-b-c)(n—2) =(>- B— + 2w)— 6/3-4y— 3t, 



(«_25-3c)(rc-2)(w-3) = 2(S-C-3a0-8£— 18£-12(7,c— 3/3-2y-/) ; 

 and substituting these in n'=a(a — 1) — 2b — 3c, and writing for n its value 

 = a(a— 1) — 23 — 3k, we have, as in the memoir, 



n'=n{n— l) 2 — rc(76 + 12c) + 46 2 +8S-f 9c 2 + 15c 

 -8£-8A-f-18/3+12y+12i-9* 



-2C-3B-30; 



viz. there is no term in w. 



Writing (n— 2){n— 3)=« + 25 + 3c + (— 4n + 6) in the equations which 

 contain (»— 2){n— 3), these become 



a(-4n+6) = 2(c — C)- a 2 ~4p -9<r-2j— 3 X - low, 

 &(-4rc + 6)=: 4& -2b 2 -9fi-C)y-3i-2p- j, 

 c(—4n + 6) = 6h —3c 2 -6(3-4 7 -2i-3a — x —3(o, 



(Salmon's equations (C)); and adding to each equation four times the cor- 

 responding equation with the factor (n — 2), these become 



a a -2a=2(a-C) + 4(k— B)-a-2/-3 x -3w, 

 26 2 -26=4/;-/3-f0y + \2t-3i-\-2p-j, 

 3c 2 -2c=6A + lO/3 + 40-2i + o<7-x + a). 



Writing in the first of these a 2 — 2a = ri + 2o+3K—a, and reducing: the 

 other two by means of the values of q, r, the equations become 



n'-a= — 2C — 4B + k-«t-2 i / — 3 x -3oi, 

 22+/3 + 3*-+;=2 P , 

 3r + c+2i + x ==5flr + / 3 + 40 + w. 

 The reciprocal of the first of these is 



f j' = a-n-t-K-2j'-3x-2C' — 4B'~ 3w'j 



