178 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
the second equation, supposing that G(m, to') is known—the third equation, supposing 
that G (to 2 , to') and G (to, to' 2 ) are known—and the fourth equation, supposing that 
G(m 3 , to'), G(m 2 , to' 2 ), G(m, m' 3 ) are known, are respectively of the form 
</> (to + to') = + <pm' + funct. (to, ml); 
and hence if a particular solution be given, the general solution is 
<f) (m) = Particular Solution + am + (3M. 
The values of the constants must in each case be determined by special considerations. 
29. The value of G (m, n, p, q) was obtained strictly; that of G (m 2 , n, p) was 
reduced to depend on G (1, 1, to 2 ), and that of G (to 3 , n) on G (1, m 3 ). I apply therefore 
the functional equations to the confirmation of the values of G (1, 1, to 2 ), G (to 2 , ft 2 ), 
and G (1, to 3 ), and to the determination of the value of G (to 4 ). 
30. First, if G(m 2 ) denote (7(1, 1, to 2 ), then G (to, to') denotes G( 1, 1, to, to'), 
which is = 2toto' ; hence 
G (to + to') 2 — G (to 2 ) — G (to' 2 ) = 2mm', 
which is satisfied by G (to 2 ) = [to] 2 . This gives 
G (1, 1, to 2 ) = [to] 2 + a to + /3M; 
but if the curve m be a system of to lines (to = to, M = 0), then G (1, 1, to 2 ) = [to] 2 ; 
and again, if the curve to be a conic (to = 2, Hi = — 1), then G (1, 1, to 2 )=1. This 
gives a = 0, /3=1, and therefore 
G (1, 1, to 2 ) = [to] 2 + M. 
31. Next, if (7 (to 2 ) denote G(m 2 , n 2 ), then G (to, to') denotes G{m,m',n®), which is 
= mm' ([?i] 2 + iV). The functional equation is 
Cr (to + to') 2 — 6r (to 2 ) — G (to' 2 ) = mm' ([w] 2 + N), 
which is satisfied by G (to 2 ) = J [to] 2 ([w] 2 + iV). Hence we have 
(r (to 2 , n 2 ) = £ [to] 2 ([w] 2 + IV) + oro + (3M, 
where a, ¡3 are functions of n, N; and observing that G(m 2 , nr) must be symmetrical 
in regard to the curves to and n, it is easy to see that we may write 
G (to 2 , nr) — \ [w] 2 [ft] 2 + M. \ [ft] 2 + N. | [to] 2 + amn + /3 (mN + nM) + yMN, 
where a, /3, 7 are absolute constants. To determine them, if the curve to be a pair 
of lines (to = 2, M = 0), then 
G (to 2 , ft 2 ) = G (1, 1, ft 2 ) = [ft] 2 + N; 
and if each of the curves to, ft be a conic (to = 2, M — —1, n — 2, N=— 1), then 
G (to 2 , ft 2 ) = 1.