523] ON THE TRANSFORMATION OF TJNICURSAL SURFACES. 391 
But we have already found 
2q + 8£ = 4 (n — 3) 6 — f ft 3 + 4ft 2 — ^n + 4, 
and we have therefore 
t — (ft — 8) 6 — ^n s + 4n 2 — + 5 — (2a,. — a>), 
and 
q = — 2 (ft — 13) & + n 3 — 14m 2 + 31ft — 18 + 4 (2a r — &>). 
I obtain these results in a different manner by investigating expressions for the 
deficiency (Geschlecht) of the nodal residue nk - 26 and for that of its projection. 
First for the projection, we have 
Twice Deficiency = [(6 — n + 4) n — 4] [(k — n + 4) n — 5] 
— 2 [(/r — n + 4) r — 1] [(& — n + 4) r — 2] (x r + 2g>, 
where I have added the term 2<o, as afterwards explained: this is 
= (k — n + 4) 2 (n 2 — 2i ,2 a } .) + (k — n + 4) (— 9n + 32ra r ) + 20 — 2 (2a,. — w), 
viz. it is 
= (k — n + 4) 2 n + (k — ii + 4) (3n — 18 — 6@) + 20 — 2 (2a,. — a>), 
or substituting for © its value — \ (n — 2) (n — 3) + b and reducing, it is 
= k 2 n +k(n 2 — 4n — 66) — 2w 3 + 16?i 2 — 32?i + 20 + 6 (n — 4) b — 2 (2a,. — &>). 
Next as regards the residue, the number li of its apparent double points is obtained 
in terms of h and t by the formula 
8 h + 6i — 2 h! = (kn — 46) (k — 1) (n — 1) — 26 (k — 1), 
(Salmon, 1. c., p. 284, except that the singularity t is not there taken account of); and 
we thence have 
Twice Deficiency = (kn — 1) (kn — 2) — 26/ 
= kn (k + n — 4) + 46 2 + 6 (— 4n — 6k + 12) + 2 — 8 h— 6t, 
or introducing q instead of h by the formula 46 2 — 86. = 4g + 46 + 241, this is 
= kn(k + n — 4) + 6 (— 4 n — 6k + 12) + 4g + 46 + 2 + 18£, 
viz. it is 
= k 2 n + k (n 2 — 4n — 6b) — 4 (?i — 4) 6 + 4g + 18i. 
So that comparing with the deficiency of the projection we have 
— 2n 3 + 16?i 2 — 32ft + 20 + 6 (ft — 4) 6 — 2 (2a,. — <w) = — 4 (ft — 4) 6 + 4</ + 2 + 18i, 
that is, 
2q + 9i = 5 (?i — 4) 6 — ft 3 + 8ft 2 — 16m + 9 — (2a,. — a>), 
the same result as before.