On Systems of Rays. 59 
cients of V7, we can deduce those of JV and T, by the methods of the foregoing 
numbers. ‘The coefficients thus found, of /V, and 7, remain unchanged through 
the whole extent of the last reflected or refracted portion of the ray, when this 
last portion is straight, the final medium being uniform ; but the coefficients of /”,, of 
the second order, change gradually in passing from one point to another, even of this 
straight portion, according to laws deducible from their connexion, already explained, 
with the constant coefficients of /7,. 
The coefficients of W, and J, of the second and higher orders, may also be cal- 
culated, whether the last medium be uniform or variable, by differentiating the expres- 
sions (V"), and eliminating the variations of o, 7, v, by the help of the conditions 
already mentioned, of maximum or minimum. 
Another method of calculating the changes produced in the partial differential 
coefficients of V7 of the second order, by reflexion or refraction, ordinary or extraor- 
dinary, into a medium uniform or variable, is to develope the second differential of 
the general formula (4"), considering AV” as a function of the seven variables «x, y, 2, 
r, y, 2, x, and considering x, y, z, as themselves functions of two independent vari- 
ables ; for example, considering z as a function of 2, y, of which the form is deter- 
mined by the equation of the reflecting or refracting surface. In this manner we 
obtain, besides the formula (W 7), which is equivalent to ten equations, the eleven 
following ; 
bar |. Bay be, Bay (&)* Bar Be 
ox* ~ Sxbz Ox O22 ox oz Ot’ 
ae: OAV Ae SAV bz m CAV (2) peek oz 
oy" dyoz dy be Ny dz 8y* ” 
Gigs CAV dz SAV & q SAL oz dz Al oz 
oxoy oxdz Oy byez «Ox oz? fx by bz) Garay’ | 
SAV SAV & SAV SAV & = 
OS Bei? Bade’ be = Qe igysen tsar Gy? Ce) 
SOA; Py ad a Lhe ADAM: Se 
~ Baby" Bedy? Be? > — Bydy * Sady’ dy? | 
on TAM RAV Be Ay | Bay & 
oxdz! zz Oa” ~ 8ydz’ | bzdz by” 
jadaY FAV Be Bar Bay & | 
oxdx ozdx or g uw oyex dzoy oy c 
which may be put under the form 
SV eV bz = 2 — Gtz 
ee : 
o=4 Sx i ~ brbz co +-(=) = ? \ (¥") 
SC 
