186 ON SKEW SURFACES, OTHERWISE SCROLLS. [339 
where 
8 — S (m 2 , n) = n ([m] + Af). 
And then putting n — 1, and comparing with the known value of NT(1, m 2 ), we find 
a = 0, /3 = — f. It will be shown, post, Art. 55, that 7 = 0, 8 = 0; and we have therefore 
cf> (m 2 , n) = nM (m — f) + N (^ [m] 2 + Af), 
and thence 
NT (m 2 , n) = i S' 2 — S + cf) (m 2 , n), 
= n (i [m] 4 + 2 [m] 3 + di ([m] 2 + m — |) + AT 2 . 
+ [?i] 2 [m] 4 + 2 [m] 3 + [w] 2 + M [m] 2 + di 2 . t) 
+ iV [m] 2 + Af). 
53. Next for (f> (m 3 ), substituting for <£ (m 2 , m') and </> (m, m-) their values, we have 
<f) (m + m') 3 — <p (m 3 ) — cf> (m' 3 ) = m!M (m — f) + M' (-1 [m] 2 + Af) 
+ mM' (m -1) + Af 4 [m'] 2 + A/'), 
which is satisfied by 
(f> (m 3 ) — M [m] 2 — |m) + Af 2 , 
and the general value then is 
<£ (m 3 ) = Af [m] 2 — fm) 4- Af 2 + am + /3Af, 
and we have 
lYT (m 3 ) =± S' 2 — S + (}> (m 3 ), 
where 
S = S {m 3 ) = ^ [m] 3 4- Af (m — 2). 
54. Taking for the curve m the (p, q) curve on the hyperboloid (m=p + q, 
M=—pq), S (m 3 ) becomes the hyperboloid taken k times, if k = [p] 3 + [^] 3 ; that is, 
S (m 3 ) = 2&, and NT (m 3 ) = 4<. ± [k] 2 + (f> (m 3 ) ; 0 (m 3 ) must vanish if p and q are each 
not greater than 3, this implies a=3, /3=11, for with these values the formula gives 
$ (m 3 ) = - i (q [p - l] 3 + p [q - l] 3 )- 
55. I assume the correctness of the value 
<f> (m 3 ) = 3m 4- df (| [m] 2 — fm 4-11) 4- d/ 2 
so obtained, as being in fact verified by means of the six several curves (1, 1), (1, 2), 
(1, 3), (2, 2), (2, 3), (3, 3); and I remark that if the foregoing value of </> (m, n, p) had 
been increased by QclMNP, then it would have been necessary to increase the value 
of <f> (m 2 , n) by 3aM 2 N, and that of <£ (m 3 ) by adf 3 ; and moreover that if the foregoing 
value of (f) (m 2 , n) had been increased by ymN + 8MN, then it would have been