232 
NOTE ON THE SKEW SURFACES 
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distance on each side from P) of the strip or element between the lines L, L x ; 
after this twisting, we imagine the strip in question to become rigid. The amount 
of the twist is such that, if dcf) be the inclination of the lines L, L x , we have 
dcf> = PQ X -7- t, viz. the twist d<f> is in a certain proportion to the shortest distance 
PQi. Similarly, keeping the line L x fixed, we twist the whole plane L 2 L 3 ... round 
PiQ 2 , the amount of the twist or inclination d<f) of the lines L x and L 2 being 
= PiQ* + Ti; after the twist, we imagine the strip or element between the two 
lines L x , L. 2 to become rigid. Proceeding in this manner, we have the series of rigid 
elements LL X , L X L 2 , L 2 L Z , &c.; putting the line L in any given position, and then 
keeping it fixed, we may turn the element LL X round this line so as to bring L x 
into a certain position; then, keeping L x fixed in this position, we may turn the 
element L X L 2 round L x so as to bring L 2 into a certain position, and so on. To 
explain this further, imagine through a point 0 a series of lines K, K x , K 2 , K 3 , &c., 
such that the inclination of K and K x is equal to that of L and L x , the inclination of 
K x and K 2 is equal to that of L x and L 2 , and so on, all these inclinations being 
otherwise arbitrary; or say that we have the double-triangle strips or elements KK X , 
K X K 2 , K 2 K Z , &c., bounded by pairs of lines, at given infinitesimal inclinations the two 
lines of a pair to each other, and forming a flexible double pyramid, which may be 
bent into any given form whatever assumed at pleasure; and this being so, we see 
that the system of the strips or elements LL X , L X L 2) L 2 L 3 , &c., may be so bent 
that the lines L, L x , L 2 , &c., shall be parallel to the lines K, K x , K 2 , &c., respectively. 
Supposing the distances PQ X , P X Q 2 , P 2 Q 3 , &c., to be all of them infinitesimal, we 
have a skew surface containing upon it a curve P X P 2 P Z , &c., which is the line of 
sfriction, viz. this is the locus of the point on a generating line which is the 
nearest point to the consecutive generating line. The line of striction cuts the 
several generating lines at an angle co, variable from line to line, which is called the 
obliquity; and the inclination between two consecutive generating lines is in a certain 
ratio to the shortest distance between the two lines. Let the inclination = shortest 
distance h- t, this magnitude r being variable from line to line: its reciprocal t -1 
is called the “ torsion ”; so that the obliquity w and the reciprocal of the torsion t 
may be regarded as functions of s, the arc of the line of striction measured from 
any fixed point. The skew surface is thus composed of rigid strips or elements, 
each included between two consecutive lines. We have further seen that the surface 
may be bent by turning these rigid elements about the successive generating lines, 
in such wise that the generating lines become parallel to the generating lines of an 
arbitrary cone, which is called the “asymptotic” cone (otherwise the director cone); 
say the surface may be bent so that it shall have a given asymptotic cone. 
I consider a given skew surface; I take x, y, z for the coordinates of a point 
on the line of striction, and a, (3, y for the cosine-inclinations of the generating line 
through this point; x, y, z, a, /3, 7 are regarded as functions of s, the length of 
or distance along the line of striction measured from any fixed point thereof; and I 
use accents to denote differentiation in regard to s. We have 
a? + /3 2 + 7 2 = 1,