110 Professor HamiLtTon’s Third Supplement 
luminous path, and therefore may assign to its cosines of direction, %z,, yz,, Z,, any 
values consistent with the first equation (B*), namely, 
2," +Yzx,” ar @7,°= 1, 
and with the following 
ats, + Bye, +2, =0, (E") 
yet when the axis of xz, has been so assumed, the perpendicular axis of y in the final 
perpendicular plane is determined, and we have 
Ty, = +(p Zr, —VYz, )s 
Yu, = + (y Tz, — 42x, ); (F") 
%y, = +(ayr, —B2,), 
the upper or lower signs being here obliged to accompany each other : and similarly 
for the initial axes of x/ and y’. 
The characteristic and related functions give immediately, by their partial differen- 
tials of the first order, the dependence of the quantities which we have denoted by 
6, T) vy) ©, 7, v, rather than that of a, B, y, a, 8, 7, on the extreme co-ordinates and 
the colour; and therefore the same functions give immediately, by their partial 
differentials of the second order, the variations 8, 87, dv, 0, é, dv, rather than 
Sa, 8B, dy, Sa, 88, Sy, in terms of dx, dy, dz, 6a’, dy’, dz, dy. But we can easily 
deduce the variations of a B ya (3 y from those of or vo'r' v' and of ayza' yz x 
by differentiating the relations 
ee ee 
Nh re ap 7 = yy? 
ov Aete EL ' ou’ 
a. Ul 
o= 3a , T= ip" = Sy ’ 
which have often been employed already in the present Supplement ; for thus we 
obtain 
= “4 + 55 2B + oe ES Se 
ht 2B + Sp oy ee id 
oe * BF share : 
on ba + Sr OB + ze fat U ee. 
ony © + 595 + on aan, 
