On Systems of Rays. 5 
in which w, w’, that is, (6% + 7° + v’) - and (6% +7r%+v")~ a are to be expressed 
ie ¢ ee =o 
as functions respectively of «(6° +7°+v°)*, r(e@++u) *, u(r t+) *a,y, 
— , "9 U " —+4 ' "2 2 19 —3 , , , 
ZX and of o (6? +77 +") cm tT (o? +7" +v") a v (eo? +77 +v") po 3 Uae oe 
After this preparation the partial differential coefficients 
sQ 8a 8 
oF Nes, Qe 
or” ou 
are homogeneous of dimension zero relatively to o, 7, v; and in like manner 
za’ 30 20 
So’ > a > Su’ ’ 
are homogeneous of dimension zero relatively to o’, 7’, v’; if, therefore, between the 
three first equations (J), we eliminate any two of the three final quantities o, 7, v, the 
third will disappear ; and similarly all the three initial quantities o’, 7’, v’', can be elimi- 
nated together, between the three last of the equations (J): and by these eliminations 
we shall be conducted to two relations of the form 
QD 
oo 
o=¥ (5, E 7, Ds %X)s 
ope ee (N) 
o= Ds E a x's ¥/, Z,x)s 
which determine the forms of the final and initial medium-functions v, v’; so that these 
forms can be deduced from the form of the characteristic function 1”. We can there- 
fore reduce to the study of this one function /’, that general problem of mathematical 
optics which has been already mentioned. 
The partial differential coefficients of the characteristic function 7, taken with 
respect to the co-ordinates 2, ¥, z, are of continual occurrence in the optical methods 
of my present and former memoirs; I have therefore thought it useful to denote 
them in this Supplement by separate symbols, o, r, v, and I shall show in a future 
number their meanings in the undulatory theory: namely, that they denote, in it, the 
components of normal slowness of propagation of a wave. 
Connection of the Characteristic Function V7, with the Formation and Integration 
of the General Equations of a Curved Ray, Ordinary or Extraordinary. 
3. It may be considered as a particular case of the foregoing general problem, to 
determine general forms for the differential equations of a curved ray, ordinary or ex- 
traordmary ; that is, to connect the general changes of direction with those of posi- 
tion, in the passage of light through a variable medium. The following forms, 
ov ov ov ov ov ov 
ay Bs a 
Sar ae sp a by 7 ay a ee © 
c 
