320 ON THE FLEXURE AND EQUILIBRIUM OF A SKEW SURFACE. [765 
G (p, q, r) = 0, where C (p, q, r) denotes a homogeneous function of (p, q, r). Hence, 
joining to the foregoing five equations this new equation 
G(p, q, r) = 0, 
these six equations determine £, 77, p, q, r as functions of a. To make the 
solution completely determinate, we have only to assume for the point P, which 
corresponds, say, to the value <r = 0, a position in space at pleasure, and to take the 
corresponding generating line PG parallel to a generating line, at pleasure, of the cone. 
As an example, writing 7 to denote an arbitrary constant angle, if the invariable 
conditions are 
I — 7, © = sin 7, II = 0, 
then the five equations are 
p 2 + q 2 + r 2 - 1 = 0, 
l 2 + m 2 + n 2 — 1 = 0, 
Ip + mq + m— cos 7 =0, 
dp 2 + dq 2 + dr 2 — sin 2 7 da 2 = 0, 
We assume first 
and secondly 
Idp + mdq + ndr = 0. 
C (p, q, r) =p 2 + q 2 —r 2 tan 2 7, = 0 ; 
C(p, q, r) = r, =0. 
Then, in the former case, we find the solution 
P, c l, r — — sin 7 sin <7, sin 7 cos a, cos 7 ; 
V> Ç — cos a, sin cr, 0 ; 
giving 
x, y, z — cos a — p sin 7 sin a, sin cr + p sin 7 cos a, cos 7 ; 
and consequently 
x 2 -by 2 —z 2 tan 2 7 = 0, 
the hyperboloid of revolution. And, in the latter case, 
p, q, r = cos (cr sin 7), sin (cr sin 7), 0, 
f, V> K = cot 7 sin (cr sin 7), — cot 7 cos (cr sin 7), a sin 7, 
that is, 
whence 
x, y = cot 7 sin z + p cos z, — cot 7 cos z + p sin z, 
« sin z — y cos z = cot 7, 
a skew helicoid generated by horizontal tangents of the cylinder x 2 + y 2 = cot 2 7. 
is a known deformation of the hyperboloid. 
This