184 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
which is 
= 2 [raw] 2 + mN + nM, 
= \ S 2 — 8 + mN + nM, 
where S—S( 1, m, n) — 2mn. 
48. And moreover 
AT(1, ra 2 ) = ND (1, ra 2 ) = ^ [ra] 4 + [ra] 3 + M (^ [ra] 2 — 2) + M 2 . \ 
-f NG (1, ra 2 ) + [ra] 3 + M (3m — 6) 
+ NR (1, ra 2 ) + | [ra] 4 + M [ra] 2 — 2m + 3), 
which is 
= \ [ra] 4 + 2 [ra] 3 + M ([ra] 2 + ra — |) + M-. 
= % S 2 — S + M(m — f), 
if' S = S (1, ra 2 ) = [ra] 2 -f M. 
The NT and NR formulae, Articles 49 to 58. 
49. I proceed to find NT (ra, n, p), &c. by a functional investigation, such as was 
employed for finding Cr(l, 1, ra 2 ), &c. Writing S(m) to denote either of the scrolls 
S(m, n, p), 8(m, n 2 ), and supposing that in place of the curve ra we have the 
aggregate of the two curves ra, m ; then the scroll S (ra + ra') breaks up into the 
scrolls 8m, 8m, and the intersection of these is part of the nodal total NT(m + m!); 
that is, we have 
NT (ra + ra') = NT (ra) + NT (ra') + 8 (ra). 8 (in'); 
and in like manner, if S (ra 2 ) stands for S (ra 2 , n), then 
NT (ra + ra') 2 = NT (ra 2 ) + NT (ra, in') + NT (ra' 2 ) + C, (8 (ra 2 ), S (ra, ra'), S (in' 2 )), 
where C 2 denotes the sum of the combinations two and two together; and so also 
NT (m + ra') 3 = NT (ra 3 ) + NT (m 2 , m') + NT(m, m' 2 ) + NT(m' 3 ) 
+ 0. 2 (S (m 3 ), S (ra 2 , in'), 8(111, ra' 2 ), 8 (ra' 3 )). 
50. Instead of assuming 
AT=££ 2 + 4>, 
it is the same thing, and it is rather more convenient, to assume 
NT = ± S 2 — S + (f>, 
viz. NT (ra) = 1 (8 (ra)) 2 — 8 (ra) + </> (ra), &c. Then observing that 
8 (ra + ra') = S (ra) + S (ra'), &c.,