339] 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
177 
25. G(m 2 , ii?).—I investigate the value by a process similar to that employed for 
G (1, 1, to 2 ). Suppose that the curves to and n are plane curves having respectively 
h and k double points; then the line of intersection of the two planes meets the 
curve to in to points, and the curve n in n points; or, combining in every manner 
the to points two and two together, and the n points two and two together, the line 
in question is to be considered as \ [to] 2 . [n] 2 coincident lines, each meeting the 
curve to twice and the curve n twice. There are besides the hk lines joining each 
double point of the curve to with each double point of the curve n. This gives 
in all | [to] 2 [?i] 2 + hk lines; or, writing h = ^ [to] 2 + M, k — [n\- + N, the number is 
= £ [to] 2 [nj + M. i [rc] 2 + N. i [to] 2 + MN; 
which is the value of G(m 2 , n 2 ) given by the investigation. 
26. G (to 3 , n).—We have 
G (to 3 , n) — n G (1, to 3 ) = n 8 (in 3 ), 
and it is in fact the same question to find G (1, to 3 ) and to find S (to 3 ). I assume 
for the present that the value [of S (to 3 ), see piost Art. 38] is = ^ [to] 3 + M (in — 2) ; and 
we then have 
G (to 3 , n) = n (i [to] 3 + M(m — 2)). 
27. Before going further, I observe that there are certain functional conditions 
which must be satisfied by the G formulae. Thus if the curve in be replaced by the 
system of the two curves in, to', instead of M we have M + M'. Let G (in) denote 
any one of the functions G (in, n, p, q), G (to, n 2 , p), G (to, n 3 ), we must have 
G (to + to') = G (to) + G (to'). 
Similarly, if G (to 2 ) denote either of the functions G (to 2 , n, p), G (in 2 , n 2 ), we must 
have 
G (to + to') 2 = G (to 2 ) + G (to, to') + G (to' 2 ) ; 
and so if G (to 3 ) stand for G (to 3 , n), then 
G (to + to') 3 = G (in 3 ) + G (in 2 , to') + G (to, to' 2 ) + G (to' 3 ) ; 
and finally 
G (m + m'y = G (to 4 ) + G (in 3 , to') + G (to 2 , to' 2 ) + G (to, in' 3 ) + G (in' 4 ). 
28. The first three equations may be at once verified by means of the above 
given values of the G functions. But conversely, at least on the assumption that 
G (to), G (to 2 ), &c., in so far as they respectively depend on the curve to, are functions 
of to and M only, we may, by the solution of the functional equations, obtain the 
values of the G functions. It is to be observed that the first equation is of the form 
(to + to') = (f> (to) + </> (to'), 
the general solution whereof is 
C. V. 
<pin — am + /331; 
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