On Systems of Rays. 121 
separating planes (VV). We might also characterise these separating planes, or 
planes of osculation, as containing the directions of mutual intersection of the same 
two touching surfaces for which V and VY’, are constant; or as the planes in which 
the deflexures of these two surfaces are equal, the ray-line at B being made the axis 
of deflexion. 
The comparison of the same two waves or action-surfaces (J?) gives a new pro- 
perty of the planes and points of transition; for the equations which determine a 
plane and point of this kind may be put under the form 
(r—r)xt+(s—s) y=, (s—s) «+(t—t)y=0, or, Sp,=Sp, oy =: (V") 
they express, therefore, that when Cis a transition point, the two surfaces (#2) touch 
one another not only at the point B, but in the whole extent of an infinitely small 
arc contained in the transition-plane. 
The point C may be called the focus of the second wave or action-surface V_, 
since all the corresponding paths of light (B’, C’) are supposed to meet in it; and in 
like manner the point 4 may be called the focus of the first surface V of the same 
kind, since all the paths (4, B’) are supposed to diverge from A. The focus A and 
the point of osculation B remaining fixed, we may change the focus C, and thereby 
the directions of osculation; but there are, in general, certain extreme or limiting 
positions for the osculating focus C, corresponding to extreme osculating waves or 
action-surfaces V, and it is easy to show that these extreme osculating foci coincide 
with the transition-points or points of vergency: and that the transition-planes or 
tangent-planes of the caustic pencils contain the directions of such extreme or limiting 
osculation. 
These theorems of intersection and osculation include several less general theorems 
of the same kind, assigned in former memoirs. It is easy also to see that they extend 
to the case when the order of the points 4 B C on a luminous path is different, so 
that B is not intermediate between A and C, and so that the paths (A, B) (A, B), 
which go from A to the points B and B, coincide at those points with the paths 
(C, B) (C, B), and not with the opposite paths (B, C) (B’, C), that is, tend from 
the point C, not to it; observing only that we must then employ the difference instead 
of the sum of the two integrals /vds, or of the two functions V and V,. 
When the point C is on a given straight ray in a given uniform medium, we can 
easily prove, by the theory of the partial differential coefficients of the second order 
of the characteristic and related functions which was explained in former numbers, 
that the equation (P) becomes quadratic with respect to z, or V, and assigns, in 
general, two or real imaginary positions C,, C,, for the transition-point, or point of 
vergency ; and that the equations (0") assign two corresponding real or imaginary 
transition-planes P, P., or tangent planes of caustic pencils. And when, besides, 
VOL. XVII. 21 
