OF THE REFLECTION AND REFRACTION OF LIGHT. 
Assuming now a quantity Ot with components £ y, £, such that 
0 \=^dt 
0t=<§> 
i=«, ^?=/3, £=y 
477^!= Fv Ot 
693 
, , ^ dX . , dr, d% 
47t/= — 4770 = — =— — , 477 li— — — 
rloi rlv. u clz ax clx ay 
and consequently 
or in terms of the components 
we may evidently write 
i. e., 
A n-ri -_ _ 
dy dz 
so that we have 
W= — .,.V, | |s( V V 0 v 0 Vjdxdyclz 
T = — ^ |) | is^dxdydz— y- (jj (t~ j r r l :j r t~)dxdydz 
Lagrange’s equations of motion may often be very conveniently represented as the 
conditions that j"(T —W)c?£ should be a minimum, or in other words that 
sj(T—W)d«=0 
and this method, from its symmetry, is particularly applicable to the methods of 
Quaternions. 
Proceeding on this method we obtain immediately the equation 
0= —[ g[| jsOiSOl.dxdT/ffe—S(Fv Sdt.cjrVv dt)dxdydz 
dt 
or in Cartesian notation 
Now we may evidently integrate the terms in SOt and S£ Sy, S£ with reference to the 
time, and the terms depending on the limits of the time must vanish separately, 
and we are not at present concerned with them, so that the equation reduces to 
jaS&SOi-f—S(Fv SO l(f)Vvdt) 
dxdydzdt =0 
or 
^(^f+^+t80+i(^.S/+^.8 9 r+fs/ l )]<M^=o 
MDCCCLXXX. 
4 U 
