S84 Professor Hamitton’s Third Supplement 
in which one of the two parameters p, g, is arbitrary, and the other depends on it by 
some assumed law; and then, for every such assumed plane locus (B"), we shall have 
to consider a partial system of the second class, deduced from and included in the 
total system of the third class (4%); namely, a system in which the equations of a 
ray-line are follows, 
ratete(S + pe) dere (S49) ays 
(c’) 
avd eee GerDe 5 
Let us therefore consider the geometrical arrangement and properties of this system 
of final ray-lines (C"), corresponding to the oblique plane locus (B") of the final 
point B’. 
The system (C'"), of ray-lines from the arbitrary oblique plane (B™), includes, as 
a particular case, the system of ray-lines from the plane of no obliquity : that is, the 
system (/”*), considered in a former number. And as the ray-lines of that particular 
system (/”*) were found to have a remarkable connexion with the guiding paraboloid 
(Z*), which touched the given perpendicular plane locus of the near final point B;, 
and which satisfied the differential condition of the second order (Y°): so, the ray- 
lines of the more general system (€”) may be shown to be connected in an analogous 
manner with the following more general paraboloid, which satisfies the same differen- 
tial condition ( Y°), and touches the more general oblique plane locus (B") at the 
given final point B, 
2=px+qytthratsry+hty; (D”) 
in which p, g, retain their recent meanings, and the coeflicients 7, s, ¢ have the follow- 
ing values, 
3 é © 
r=— (2+ pS); t=— (Bap rE) : 
é 
4(E +5 3 +3+ z = t 93 a: 
But in order to develope this more general connexion, between the ray-lines ( C”), and 
the paraboloid (D”), it will be useful previously to establish some general theorems 
respecting the deflexures of curved surfaces, which include some of the known theo- 
rems respecting their curvatures and planes of curvature. 
Let us then consider the paraboloid (D"), or any other curved surface which has, 
at the origin of co-ordinates, a complete contact of the second order therewith, and 
which is therefore approximately represented by the same equation: that is, (on account 
of the arbitrary position of the origin, and arbitrary values of the coefficients p, 9,7,5,¢,) 
any surface of continuous curvature, near any assumed point upon this surface. The 
tangent plane at this arbitrary point or origin, has for equation 
(E") 
s= — 
