On Systems of Rays. 51 
i, like v, being here treated as a function of o, 7, 7, y, z, x: and if we put, as usual, 
ye) 62 re 
S0=808 + ara & — ae au & — oy ara 2 + apm + 8208 iw + ee , (V*) 
and oe 
= 303 a ae + aro e+ aye + 8 + ea (W") 
we find 
OS OE) 
Se bu ue to a ee 
+ 28. (80 — gor —5 gor — 5 by 5 be Fx) 
2 8 76 
— 8 (oe + Sor ge + 52) (We te Be d» &) 
in which 
bof ae: Pig ce 5 
sd=r? (a +32 aoe pte Ox aah to GED 
and which is equivalent to twenty-eight expressions for the partial differential coefti- 
cients of Q of the second order: it gives, for example, 
oOU Q eu 20U 
sO 302 oT 5 Tete - 
Bet fe cial se Ag Se ony 
oc 8 
And since the forms of the connected functions Q, v, v, of which each expresses 
the optical properties of the final medium, may be deduced, by the method of the 
second number, from the form of the characteristic function /’, it evident that their 
partial differential coefficients also, of all orders, are not only related to each other, 
but may be deduced from the coefficients of that one characteristic function. 
General Formula for Reflection or Refraction, Ordinary or Extraordinary. Changes 
of V,W,T. The Difference AV is =0; AFV=AT= a Homogeneous Func- 
tion of the First Dimension of the Differences Ao, Ar, Av, depending on the 
Shape and Position of the Reflecting or Refracting Surface. Theorem of Mas- 
ma and Minima, for the Elimination of the Incident Variables. Combinations 
of Reflectors or Refractors. Compound and Component Combinations. 
11. Let us now endeavour to improve our theory of the characteristic and related 
functions, by applying the methods of the present memoir to improve the determina- 
