[339 
339] ON SKEW SURFACES, OTHERWISE SCROLLS. 185 
! 1 
• 2 
the foregoing equations for AT give 
cf> (to + to') — cf> (m ) + cf> (in), 
(f> (to + to') 2 = <f> (to 2 ) + <f> (to, in') + 0 (to' 2 ), 
(f) (to + to') 3 = (j) (to 3 ) + <f> (to 2 , to') + 0 (to, to' 2 ) + cf> (to' 3 ); 
and if in the second equation <f> (to, in') and in the third equation (in-, in') and 
</> (to, to' 2 ) are regarded as known, these are all of them of the form 
/(to + to') — /(to) — /(to') = Funct. (to, to') ; 
so that, a particular solution being obtained, the general solution is f(m)~ Particular 
Solution + am + /3ilf, at least on the assumption that / (to), in so far as it depends 
on the curve to, is a function of to and M only. 
a, such as was 
of the scrolls 
we have the 
1 up into the 
[ NT (m + m'); 
51. First, if cf) (in) stands for <£ (to, n, p), we obtain </> (to, w, p) = am + f3M, or 
observing that </> (to, ?i, p) must be symmetrical in regard to the curves to, w, and p, 
we may write 
(j) (to, w, p) = amnp + ¡3 (Mnp + Nmp + Pmn) + 7 (mNP + 11MP + pMN) + S MNP, 
and then 
NT (to, 11, p) — ^S 3 —8+(f) (to, 11, p), 
= 2mnp (mnp — 1) + cf) (to, 11, p). 
But for p = l this should reduce itself to the known value of iVT(l, to, ?i); this gives 
a = 0, ¡3 = 1, 7 = 0; we in fact have, as will be shown, post, Art. 55, 3 = 0; and hence 
NT (to, 11, p) = £ S 2 — S + (Mnp + Nmp + Pmn), 
= 2 [mnp] 2 + (Mnp + Nmp + Pmn). 
S (»»'*)), 
52. Next, if </> (to 2 ) stand for (/> (to 2 , 11), then cf) (to, to') stands for <f>(m, to', n), 
which is = Nmm' + n (mM' + m'M), and the equation is 
so also 
<fi (to + to') 2 — cf) (to 2 ) — </> (in' 2 ) = Nmm' + n (toM' + m'M). 
A particular solution is </> (to 2 ) = 1 [to] 2 A + ?ztoA, and we have therefore 
<£ (to 2 , n) = i [to] 2 N + nmM + am + ¡3M; 
or observing that 0 (to 2 , w) considered as a function of n, satisfies the equation 
(f) (n + n') = (f> (n) + (f) (n'), 
and is therefore a linear function of n and N, we may write 
cf) (to 2 , n) = i [to] 2 N + 11mM + anm + (3nM + 7mN + SMN; 
we then have 
NT (m 2 , n) =) S 2 — S + (p (in 2 , 11), 
C. y. 
NT (to 2 , n) 
24