On Systems of Rays. 113 
straight rays, corresponding to any given luminous or initial point 4, and to any 
given colour x, two real or imaginary points of vergency B,, B,, on any one straight 
final ray, that is, two points in which this ray is intersected by infinitely near rays of 
the same final system ; and the joint equation in w y z, (involving also a’ 7/ 2’ y as 
parameters, ) of the two caustic surfaces which are touched by all the final rays and 
are the loci of the points of vergency, may be obtained by eliminating o + v between 
the equations (//”*) and the quadratic ("): which quadratic, by the homogeneity 
of the functions JV and Q +1, may be put under the following simpler form, 
(ou =) (ee ae) Z ae xO ne A 
get! xe) oe + ge) —\eae +” sc5 (M") 
and admits of several other transformations. When V” has either of the two values 
determined by this quadratic, that is, when the final point B of the luminous path has 
any position B, or B, on either of the two caustic surfaces, then the equations 
deduced from (/V’*) by differentiating with respect to x y 2 as well as oz v, namely, 
ae poe Sern ae) U4 
or — > (ode +7dy +v0z) = 8 3, + V3~ ; 
Saigae wie ow 8a 
ey =o (coax + 7réy + vdz) = ae Be ais ’ (N") 
Bye) : AY Aye) 
8z Se (oot + rey +vdz) = os + Vs~ ; 
conduct to a linear relation between da, 8y, 82, which may be put under several forms, 
for example under the following, 
1 6Q 1 6Q 
X fee -F tater rita) bar dy ae (ode + rly +82) b 
=" fe = (oda + dy +82) } > (O*) 
in which we may assign to d d' X” any of the following systems of values, 
ow So 6, SV aso 5, Sw “0 
LES Ss “A Aa ek er a rE 
ao ee Pp BO ye Oy i 
Second A= Sear t Ve » A=, + V se? * =z3, + V ss, a CE) 
eM ie: 8M a BO BM PO 
fi SEPT aati i 7 8rieatad odelat. T Balae So 
and it is easy to see that the linear relation thus deduced, between 6a, dy, dz, is the 
differential equation, or equation of the tangent plane, of the caustic surface at the 
point of vergency x y 2. The same linear equation represents also the plane of 
VOL. XVII. 26 
