790] 
GEOMETRY. 
573 
34. In the general case of a singly infinite system of lines, the locus is a ruled 
surface (or regulus). If the system be such that a line does not intersect the con 
secutive line, then the surface is a skew surface, or scroll ; but if it be such that 
each line intersects the consecutive line, then it is a developable, or torse. 
Suppose, for instance, that the equations of a line (depending on the variable 
parameter 0) are 
' + -=0 1 + 
- - - = - ( 1 _ ^ 
then, eliminating 0, we have ^ — Z — = 1 — , or say ^ + j- 2 — — 2 = 1, the equation of a 
CL C" 0 CL 0 C 
quadric surface, afterwards called the hyperboloid of one sheet; this surface is con 
sequently a scroll. It is to be remarked that we have upon the surface a second 
singly infinite series of lines; the equations of a line of this second system (depending 
on the variable parameter <f) are 
It is easily shown that any line of the one system intersects every line of the other 
system. 
Considering any curve (of double curvature) whatever, the tangent lines of the 
curve form a singly infinite system of lines, each line intersecting the consecutive line 
of the system,—that is, they form a developable, or torse; the curve and torse are 
thus inseparably connected together, forming a single geometrical figure. A plane 
through three consecutive points of the curve (or osculating plane of the curve) 
contains two consecutive tangents, that is, two consecutive lines of the torse, and is 
thus a tangent plane of the torse along a generating line. 
Transformation of Coordinates. 
35. There is no difficulty in changing the origin, and it is for brevity assumed 
that the origin remains unaltered. We have, then, two sets of rectangular axes, 
Ox, Oy, Oz, and Ox\, Oy x , 0z 1} the mutual cosine-inclinations being shown by the 
diagram— 
X 
y 
Z 
iCj 
a 
P 
y 
2/1 
a 
P' 
y 
«1 
n 
a 
P" 
tr 
y 
that is, a, ¡3, 7 are the cosine-inclinations of Ox 1 to Ox, Oy, Oz; a, /3', f those of 
Oy u &c.