56 Professor Hamitton’s Third Supplement 
can deduce from it, by the methods of the fourth number, the other auxiliary func- 
tion JV, and the characteristic function V. 
In general for all optical combinations, whether with uniform or with variable 
media, we have, by the definitions of the functions /, VY, 7, and by the results of 
former numbers, the following expressions, 
i ov ov ; 
V=f? vds; T= f"'(#5+95,+ 2) ds ; 
a4 ER i) s ov ov _ ov ; 
We=x0+y7 +z +f (*Et 957 z=) ds : 
(Q’) 
ds being, as before, the element of the curved or polygon ray ; and hence it follows 
that if we consider any total combination, of m+—1 media, whether uniform or 
variable, as resulting from two partial combinations, of m and of x media respectively, 
combined so that the last medium of the one partial combination (7) is the first of the 
other partial combination (7), and so that the final rays of the one partial combination 
are the initial rays of the other, then the functions V7, 7, (but not in general JV) for 
the total combination, are the sums of the corresponding functions for the partial com- 
binations : it follows also from the general expressions for the variations of these func- 
tions, that the intermediate variables, belonging to the last medium of the first partial 
combination, or to the first medium of the second, are to be eliminated from the sum, 
by the condition of making that sum a maximum or minimum with respect to them, 
Analogous remarks apply to compound combinations, composed of more than two 
component combinations. These properties of the functions V, T, for total or result- 
ant combinations, will be found useful in the theory of double and triple object-glasses, 
and other compound optical instruments. 
Changes of the Coefficients of the Second Order, of V, WW, T, produced by 
Reflexion or Refraction. 
12. With respect to the changes produced by reflexion or refraction in the coefti- 
cients of the second order, of the characteristic function /”, and therefore also of the | 
connected functions W, T, they may be deduced from the following formula, analo- 
gous to (C”), 
CAV =8. UW =dANU + VWACU 5 (R’) 
u, \, having the same meanings as in (B’) (C’); and the multiplier \, which was 
introduced also in the First Supplement, and was there regarded as a function of the 
final co-ordinates x, ¥, %, beg now considered as involving also the initial co-ordi- 
nates ’, y', 2, and the chromatic index x- The seven variations dr, dy, dz, da’, dy’, 
