765] ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. 319 
points g, g', t respectively, we have a spherical triangle gg't, the sides of which are 
I — ildcr, I, and ©do-, and of which the perpendicular g'm is = Ada ; we have thus 
an infinitesimal right-angled triangle, the base and altitude of which are flda, A da, 
9' 
0^ 
9 Qda- rn g 
and the hypothenuse is <dda; whence © 2 = il 2 + A 2 . In the case of the developable 
surface A = 0 and © = il. It may be remarked that, when the curve on the skew 
surface is the line of striction, Ave have 0=0; in fact, taking P to be on the line 
of striction, the line 
X-f = Y-v = 
qr — q'r rp' — r'p pq' — p'q ’ 
through (£, 11, £) at right angles to the two generating lines, meets the consecutive 
generating line X, Y, Z = £' + pp', rj' + pq', £' + pr'; and the condition that this may 
be so is easily found to be il = 0. 
Take, for a moment, p', q', r' for the cos-inclinations of the consecutive generating 
line P'G'; we have 
Ip + mq + nr = cos /, 
Ip' + mq + nr' = cos (/ — Clda), 
pp' + qq' + rr' = cos %da; 
and then writing p', q', r =p + dp, q + dq, r + dr, and observing that the equation 
p' 2 + q 2 + r' 2 = 1 gives 
pdp + qdq + rdr = — \ {dp 2 + dq 2 + dr 2 ), 
these equations and the before-mentioned two equations become 
(Z7j) p 2 + q 2 + r 2 — 1 = 0, 
( U 2 ) l 2 + m 2 + n 2 — 1 = 0, 
( U 3 ) Ip + mq + nr — cos 1 = 0, 
( U 4 ) Idp + mdq + ndr — il sin Ida = 0, 
( U s ) dp 2 + dq 2 + dr 2 — S 2 da 2 — 0, 
which equations, considering therein l, m, n as standing for their values ^, 
are the geometrical relations which connect the six variables rj, p, q, r, considered 
as functions of a. And in these equations I, il, © denote given functions of a, 
invariable by any flexure of the surface. 
To complete the geometrical theory, it is to be observed that we can by flexure 
bring the generating lines of the surface to be parallel to those of any given cone