94 Professor Hamitton’s Third Supplement 
determination of a surface of the second degree, if the two points in which the para- 
bola( 7") is intersected by the axis of y, that is, by the intersection-line of the planes 
of the two curves, namely, the origin and the point 
o—0; —— 3=0p (K") 
were not also contained on the ellipse or hyperbola (S'°). But we may confine our- 
selves to the last chosen conditions, of having these two known curves as the inter- 
sections of the hyperboloid with two known planes, and of having known directions 
for the asymptotes of its hyperbolic curve of intersection with a third known plane, 
as adequate and sufficiently simple conditions for the construction of the sought 
hyperboloid, and thereby of the auxiliary paraboloid (D"), to which that hyperboloid 
osculates. And with respect to the new constant of deviation m, connected with this 
auxiliary paraboloid, we may put its general yalue (#"") under the form 
n=n,+hpe — 49%, (L") 
n, being the particular value 
op 8a 
=e 14 
eee (3. ay) an 
for the plane of no obliquity, that is, the value (6") connected with the guiding para- 
boloid (Z*) in the theory of the elements of arrangement which was given in a former 
number : we may therefore construct the new constant 7, as the ordinate z of a plane 
Z=px+qytn,, (N™) 
which is parallel to the given oblique plane (B”), and contains the point 
z=0, y=0, z=n,, (O") 
so that it intersects the axis of z at a distance from the origin =the old constant of 
deviation m,. The other co-ordinates x, y, to which the ordinate z = 7 corresponds, 
are 
e=42, y=-3e (P") 
so that the corresponding line / 2° +7’ is equal to half the curvature of the ray, and 
is perpendicular to the radius of that curvature. 
The details of the present number have been given, in order to illustrate the sub- 
ject, by combining it more closely with geometrical conceptions; but the new auxiliary 
paraboloid, and the new constant of deviation, might have been considered as suffi- 
ciently defined by their former algebraical expressions. 
