On Systems of Rays. 109 
OL =NXe, 84, +4y, Sy, +a 82, 
SY =Yr, 8%, +Yy, Sy, +B 8z,, 
Sz=z, da, +2 dy,+y (A") 
Sa =2'n/ 8a) +a'y: Sy’ +a 82", 
by =y'r) 8; +4; By, +B 8z/, 
82 =2'z' 8a) +2y) Sy) +7 8z;, 
because 
Bz =a, Yx, =P, 2s, =75 Be 
we=a, y'x=P, Z2;=7 5 oe 
we have also 
a —On0,—0; La ig °Y, =O} (c%) 
a, =0, B,=0, y,=1, Sy, =0, 
and therefore, by (£’*), 
Sa =az 8a,+2y, 88,3 8a =a'n/ 8a) +2'y/ 98/5 
SB=yr, 8a, +yy, 88,5 BB =y'x; ba) +r; 83/5 (D") 
Sy=22, 8a,+2y, 88,3; Sy =2'r/ 8a, +2y/ 8B; 
and substituting these values (4") (D") for the twelve variations 62, dy, dz, 6x’, dy’, dz, 
8a, 8, Sy, 8a’, 38, Sy’, in the general linear relations (4°) between these twelve varia- 
tions and the variation of colour 8y, or in any other linear relations of the same kind, 
deduced from the characteristic and related functions, and referred to arbitrary rec- 
tangular co-ordinates, we shall easily discover the particular dependence, of the form 
(D*), of 8a,, 88,, on dx, dy,, 8z,, da’, dy/, 8x, and of da/, §3/, on dx, dy,, dx, dy/, 
oz), ox. 
We seem, by this transformation, to introduce twelve arbitrary cosines or coefii- 
cients, namely, 
*’ LA s / , ‘y , ‘ ’ / se 
Ur,5 Yr,» Fx,5 Vy,» Yy,> %y,» Vas Y v/s Fay Vy/s Yy/s Zy/ 5 
but these twelve coefficients are connected by ten relations, arising from the rectangu- 
larity of each of the four sets of co-ordinates, and from the given directions of the 
semiaxes of z and z’; so that there remain only two arbitrary quantities, correspond- 
ing to the arbitrary planes of x, z,, x) z/, of which planes we often, lately, disposed 
at pleasure, so as to make them coincide with certain given planes of curvature, or 
otherwise to simplify the recent geometrical discussions. Thus, although we may 
assign to the semiaxis of xv, any position in the given final plane perpendicular to the 
VOL. XVII. 2F 
