339] ON SKEW SURFACES, OTHERWISE SCROLLS. 179 
These cases give a = ¡3 = 0, y = l, and therefore 
G (to 2 , n") = % [to] 2 \nf + if. [77] 2 + N. [to] 2 + MN. 
32. Again, G(m s ) standing for G (1, to 3 ), then G (to 2 , to') and Cr (to, to' 2 ) will stand 
for Cr (1, to 2 , to') and Cr (1, to, to' 2 ), the values whereof are to'([to] 2 + if) and to ([??z'] 2 + if') 
respectively. We have thus 
G (to + to') 3 — Cr (to 3 ) — Cr (to' 3 ) = on ([to] 2 + if) + in ([to'] 2 + if'), 
a solution of which is G (to 3 ) = | [?w] 3 + mM. Hence we have 
G (1, to 3 ) = 4 [to] 3 4- TOif + aw + /3M. 
Suppose first that the curve to is a system of lines (m — om, if = 0), then G( 1, to 3 ) = ^ [to] 3 ; 
and next that the curve to is a cubic in space or skew cubic (to = 3, if — — 2), then 
Cr(l, to 3 ) = 0, since a line can meet the curve in two points only. We thus find a = 0. 
& =— 2, and thence 
G (1, to 3 ) = ^ [to] 3 + if (?n. — 2). 
33. Hence, substituting for G (to 3 , to'), Cr (to 2 , to' 2 ), Cr (to, to' 3 ) their values 
to'(t [to] 3 + if (to — 2)), ]r [??i] 2 [??i'] 2 4- if. [to'] 2 + if'. ^ [to] 2 + MM', and to (^[to'] 3 + if' (to' — 2)) 
respectively, we find 
G (to + to') 4 — Cr (??i 4 ) — G (oil 4 ) = on' [to] 3 + if (to — 2)) 
+ ^ [7n~\ 2 [to'] 2 + if. I [on'] 2 + if' . ^ [to] 2 + ifif' 
+ to (} [to'] 3 + if' (to' — 2)), 
and thence, obtaining first a particular solution, the general solution is 
G (to 4 ) = ytj \on] 4 + if (^ [to] 2 — 27n) + if 2 . -| + OLVl + /3if. 
34. To determine the constants, suppose first that the curve on is a system of 
lines (to = to, if = 0), we must have G (on 4 ) = [??f| 4 + on, and thence a = 0. Next, if 
the curve to be a conic (on = 2, if = — 1), we must have Cr (to 4 ) = 0; and this gives 
/3 = JJ-, and consequently 
G (on 4 ) = jL [to] 4 + 777 + if (£ [to] 2 — 2777 + - 4 2 l ) + if 2 . 
The NG formulae, Article 35. 
35. The NG formulae are now at once obtained, viz. we have 
NG (on, 01, p) = G (on 2 , 7i, p) + G (on, n 2 , p) + G (on, 01, p 2 ), 
NG(m 2 , n) = SG(m 3 , 00) + G (on 1 , n 2 ), 
NG (on 3 ) = 6G (on 3 ), 
which give the values in the Table. 
23—2