On Systems of Rays. 111 
5, ; referring; as in former numbers, to the variations of x, y, z, x, and 8’ to those of 
a’, y', %, x: and hence we have, by some symmetric eliminations, 
Ray Mer Es ae oy oe e-( B85) 
v'da = (2 yy 
op ~& Sal ~ Sad “ §B/ ~ dy8a “Sy 
v'§8 = Gr +e) ( er —8, 55) eat sy—8, ys Lay 28,2"), 
vty = (Sr 4R) (H- =8,5) yp ( 8,2) aap (ar-2 a 
vids (Fee + Se) ( 86-8 57) ~arag( B= 2 ia) “1 
was (Sa + Fe) ( BB 5) ap B85) grag Be —8 S), 
vy = ot + 5gu (2-2 oe )- — sg (et —8 oY a say (8! =3) =), 
v' having the iy (L*), and v"" ine erplogaus meaning 
Sv! 2 So Se’ ow’ 2 
Sea ORE RONAN | (aa ys 
= ha? = a + 3p" = <(ans my) + a we Gerd OY) 
We ae also deduce the variations of a B y a’ B’ y' from those of ctuad ru Hy z 
x’ y' z x, by differentiating the equations (J) of the second number, and by employ- 
ing the functions Q, ©’, instead of v, v 
(H"’) 
The general Linear Expressions for the Arrangement of Near Rays, fail at a 
Point of Vergency. Determination of these Points, and of their Loci, the 
Caustic Surfaces, in a Straight or Curved System, by the Methods of the present 
Supplement. 
23. We have hitherto supposed that the infinitesimal or limiting expressions of the 
variations of the extreme cosines of direction of a luminous path, are linear functions 
of the variations of the extreme co-ordinates and colour. But although this supposi- 
tion is in general true, it admits of an important and extensive exception ; for the 
linear form becomes inapplicable when the given luminous path (4, B),, with which 
other near paths are to be compared, is intersected in its initial and final points 4, B, 
by another path infinitely near, and having the same colour y: since then the extreme 
directions may undergo certain infinitesimal variations, while the extreme positions 
A, B, and the colour x, remain unaltered. It is therefore an important general 
problem of mathematical optics, to determine, for any proposed optical combination, 
the relations between the extreme co-ordinates and the colour of a luminous path 
