On Systems of Rays. 61 
Let us also establish, according to the analogy of our former notation, the following 
definitions similar to (P), 
dx, oe dy, es 
i LRA a Aptana aa a cs 
ee 8! dy; dz} (C’) 
=== =— = —+ 
, itn ales de”? ds’ ” 
and the following, similar to (), 
OOK 8K Eincl 
Lie? Ts Bah leah, 8%) 2 
y Y F (D‘) 
’ oV epee Oe. 
o, 5 i ) 1 oy; ee 32’ 
we shall then have 
ata vy +B, Vy Fy) Uz 5 | 
B=a, Ye, +2, Yy, +, Yz,3 
Y=4, 72, +B, zy, HY, %2,3 (E’) 
a = a, Lig! a B; Ly! as Y, X's! > 
U ‘ ' a , é ‘ 
B =a; Yo +B, Wy +7, Yes 
y = a’ Zn! a5 ‘ow By! ate y, Bie! > 
and 
/ Ul , | Pas | (i } 
o,=o08z, + TY x, SF Vex, 5 6, =oX x! +TY 2 tue x5 
T,=OLy, +TYy, + ey, 5 T= omy’ +7 y'y/ + vey! 3 (EY) 
v=o, +TYz, tues, 3 v, =o 02" + TY 2/ 5 UZ a! - 
And if, by substituting in the former homogeneous medium-functions, v, v', the ex- 
pressions (Z*) for a, 8, y, a, 8’, y, we obtain v under a new form, as a homogeneous 
function of a,, 3, y,, of the first dimension, and v' as a homogeneous function of the 
same dimension of a‘, 8’, y/, and then differentiate these new forms of v, v’, with 
reference to their new variables, we find, by (4°), the following relations between the 
new and the old coefficients, 
ov Ov ou ov 
Sf be + 3p a2 By 3 
ov ov ov ov 
‘Yeh wna Y, + + By 3 
ov bv ou 
(G*) 
ee ee ae af aides 2! re 
§B/ SH Y, ae yew Yy/ + by ¥,> 
ov eu’ ie wr, Fa Soe, 
Sear be baa Yet Hy 8S 
oy oa” 8G Y 2; eae 
VOL. XVII. 
| 
DeGmOu A, By ou’ | 
R 
