322 
ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. [765 
and the second line, attending in like manner only to the terms under the sign S, 
gives 
The fifth term, written under the form 
and attending only to the terms under the sign S, gives 
where in each case I have marked with an asterisk the lines which present them 
selves in the final result. 
Hence, joining to the foregoing the force-terms 
X8Ç + Y8 V + Z8Ç + L8p 4- M8q + Mr, (*) 
and equating to zero the coefficients of 8%, 8v, 8Ç, 8p, 8q, 8r respectively, we have 
( 
0 = X 
0=7 
d rp d d dr 
-L oil V- loV 
0 = Z 
d m j d m dp 
da 4 ~Ta 5 da’ 
0 = L +T lP 
+ T,l 
0 = M+T,q 
+ T 3 m 
0 = N + T x r 
\ 
d% dr] dÇ 
da’ da’ da’ 
, the variables being 
where it will be recollected that l, m, n stand for 
f, rj, p, q, r, and a. The elimination of T 1} T 2 , T 5 from the six equations 
should lead to a relation between £, r\, p, q, r, which, with the foregoing five 
relations, would determine the six variables rj, £, p, q, r in terms of a. 
In particular, the forces and moments X, Y, Z, L, M, N may all of them 
vanish; assuming that T 1} T 2 , ..., T 5 do not all of them vanish, we still have the 
sixth relation, which (with the foregoing five relations) determines f, r], p, q, r in 
terms of a; and it is to be remarked that the problem in question, of the figure 
of equilibrium of the skew surface not acted upon by any forces, is analogous to 
that of the geodesic line in space; only whilst here the solution is, curve a straight 
line, the solution for the case of the skew surface depends upon equations of a 
complex enough form; in the case of the developable surface, the required figure is 
of course the plane.