318 ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. [765 
The summation S extends first to the different points of the generating line, and 
then to the different generating lines; applying it first to the particular generating 
line, we write 
SX'dm, SY'dm, SZ'dm, SX'pdm, SY'pdm, SZ'pdm 
= X, Y, Z, L, M, X, 
where X, Y, Z are the whole forces, and L, M, N the whole moments about the 
point P, for the generating line 6r; retaining the same summatory symbol S, as now 
referring to the different generating lines, the equation becomes 
S{X8g+YSv + ZS£+L8p + M8q + №r+T 1 8U 1 + ... + T 5 8U 5 } = 0. 
We have now to consider the geometrical theory of the flexure. Taking on the 
skew surface an arbitrary curve cutting each generating line G in a point P, coordinates 
(I, ij, £), and taking cr for the distance along the curve of the point P from a fixed 
point of the curve ; also p, q, r, as before, for the cos-inclinations of the generating 
line G, then when the surface is in a determinate state, £, rj, £, p, q, r are given 
functions of cr; but these functions vary with the flexure of the surface, with, however, 
certain relations unaffected by the flexure ; and the problem is to find first these 
relations. As already mentioned, one of them is p 2 + q 2 + r 2 — 1 = 0. 
Taking P' as the consecutive point on the curve, so that the direction of the 
element PP' is that of the tangent PT at P, it is convenient to write l, m, n for 
the cosine-inclinations of the tangent; we have, it is clear, 
l, m, ri 
di% dr] dÇ 
da ’ da ’ da ’ 
l 2 + to 2 + n 2 — 1 = 0. 
The conditions in order to the rigidity of the strip, are that the angles GPP, 
GPP (= 180° — GP'T), and the inclination GP' to GP, shall have given values, 
variable it may be from strip to strip—that is, these values must be given functions 
of a. Taking Z GPT = I, the value of GP'T can differ only infinitesimally from that 
of GPT, and we take it to be G'P'T = I — if da; also the inclination GP to GP' 
is an infinitesimal, = ®da: we have I, if, ® given functions of a. It is to be 
remarked that these conditions imply, inclination of GP' to tangent plane GPT at P 
has a given value Ada; in fact, if through P we draw a line P7 parallel to P'G', 
then, if P is regarded as the centre of a sphere which meets PG, P<y, PT in the