309 
the object and the image by the methods used in the geometry 
of homographic figures. The whole theory of images formed 
by simple or compound instruments when aberration is not 
considered is thus reduced to simple proportion, and this is 
found very convenient in the practical work of arranging lenses 
for an experiment, in order to produce a given effect. 
As a preparation for physical optics the same elementary 
problems may be treated by Hamilton’s method of the Charac- 
teristic Function. This function expresses, in terms of the 
coordinates of two points, the time taken by light in travelling 
from the one to the other, or more accurately the distance 
through which light would travel m a vacuum during this 
time, which we may call the reduced path of the light between ~ 
the two points. The relation between this reduced path and 
the quantity which occurs in Cotes’ celebrated but little 
known theorem, is called by Dr Smith the “apparent distance.” 
The relations between the “apparent distance” and the posi- 
tions of the foci conjugate to the two points, the principal foci 
and the principal focal lengths, were explained; and the general 
form of the characteristic function for a narrow pencil in the 
plane of ar was shewn to be 
Vea Vo pyri t bets 
a lp, (ge a) a, + Me Ge a,) a, aha (fits + Sots) U,V, + &e, 
2 (7, a a.) (r, ae a.) Swiss : 
where 7,, 7, are measured from the instrument in opposite 
directions along the axis of the pencil in the media /,, My: 
respectively, and x,, x, are perpendicular to the axis. 
a,, % are the values of , 7, for the principal foci, and 
f> fy the principal focal lengths, and fw, =f, 
Tf ae ic aeres : 
uP T,—d, 2%’ 
the last term of V assumes the form = and an infinite number 
