580 
GEOMETRY. 
[790 
The tangent is the line through the point (x, y, z) and the consecutive point 
(x + dx, y+ dy, z + dz); its equations therefore are 
£ — X 7] — y £ — Z 
dx dy dz 
The osculating plane is the plane through the point and two consecutive points, 
and contains therefore the tangent; its equation is 
or, what is the same thing, 
%-x, 
y-y> 
Ï-* 
dx , 
dy , 
dz 
d-x, 
> 
d' 2 z 
(£ — x ) (dyd?z — dzd?y) + (y — y) {dzd 2 x — dxd-z) + (£ — z) (dxd 2 y — dyd 2 x) = 0. 
The normal plane is the plane through the point at right angles to the tangent. 
It meets the osculating plane in a line called the principal normal; and drawing 
through the point a line at right angles to the osculating plane, this is called the 
binormal. We have thus at the point a set of three rectangular axes—the tangent, 
the principal normal, and the binormal. 
We have through the point and three consecutive points a sphere of spherical 
curvature,—the centre and radius thereof being the centre, and radius, of spherical 
curvature. The sphere is met by the osculating plane in the circle of absolute 
curvature,—the centre and radius thereof being the centre, and radius, of absolute 
curvature. The centre of absolute curvature is also the intersection of the principal 
normal by the normal plane at the consecutive point. 
Surfaces; Tangent Lines and Plane, Curvature, &c. 
39. It will be convenient to consider the surface as given by an equation 
f {%, y, z) = 0 between the coordinates; taking (x, y, z) for the coordinates of a given 
point, and (x + dx, y + dy, z -+- dz) for those of a consecutive point, the increments 
dx, dy, dz satisfy the condition 
df 
dx 
dx + < ~dy + 
df 
dz 
dz = 0, 
but the ratio of two of the increments, suppose dx : dy, may be regarded as arbitrary. 
Only a part of the analytical formulae will be given ; in them £, y, £ are used as 
current coordinates. 
We have through the point a singly infinite series of right lines, each meeting 
the surface in a consecutive point, or say having each of them two-point intersection 
with the surface. These lines lie all of them in a plane which is the tangent plane; 
its equation is 
as is at once verified by observing that this equation is satisfied (irrespectively of 
the value of dx : dy) on writing therein f, v , £ = x + dx, y + dy, z + dz.