UPON THE PROPERTIES OF LIGHT. 
241 
If, however, we would take the forces more correctly (fig. 10), let A and B be the two 
edges, and let their spheres of flexion be equal, AC(=a) being A’s sphere of inflexion 
and B’s sphere of deflexion ; BC (=a) being A’s sphere of deflexion and B’s sphere of 
inflexion ; and let the flexion in each case be inversely as the mth power of the di- 
stance. Let CP=.i’, PM=i/, the force acting on a ray at the distance a-\-x from A 
and a—x from B. Then if B is removed and only A acts, y= acts, 
t/=— ^ 
“ [a + x)”' ' {a—x)”'" 
Now the loci of y andy are different curves, one similar to a conic hyperbola, the 
other similar to a cubic; but of some such form when m=\, as SS' and TT'. It is 
evident that the proportion of y \y' can never be the same at any two points, and 
consequently that the breadths of the fringes made by the action of one can never 
bear the same proportion to the breadths of those made by the action of both, unless 
we introduce some other power as an element in the equation, some power whereby 
from both values, 3/ and y, x may disappear, else any given proportion of y \y' can 
only exist at some one value of x. Thus suppose (which the fact is) y \y' wX '.h 
or 1 : 6, say : : 1 : 6, this proportion could only hold when 
(5™— l)a (4™ — l)a .f \ y r 
x= > ■ or = > if 3/ :y : : 1 : 5 . 
5 "* + 1 4 ”* + 1 
When m=2, the force being inversely as the square of the distance, then x=- x and 
(V 5 - 1 ) 
\^5 + l 
a, are the values at which alone y:y'::l :5 and 1 : 6 respectively. 
But this is wholly inconsistent with all the experiments ; for all of these give 
nearly the same proportion of y:y' without regard to the distance, consequently the 
new element must be introduced to reconcile this fact. Thus we can easily suppose 
the conditions, disposition and polarization (I use the latter term merely because the 
effect of the first edge resembles polarization, and I use it without giving any opinion 
as to its identity), to satisfy the equation by introducing into the value of 3/ some func- 
tion of a ~x. But that the joint action of the two edges never can account for the 
difference produced on the fringes, is manifest from hence, that whatever value we 
give to m, we find the proportion of y' -.y when jc=0 only that of double, whereas h 
or 6 times is the fact. The same reasoning holds in the case of the spheres of flexion 
being of different extent; and there are other arguments arising from the analysis 
on this head, which it would be superfluous to go through, because what is delivered 
above enables any one to pursue the subject. The demonstration also holds if we 
suppose the deflective force to act as^ of the distance, while that of inflexion acts as^^» 
But I have taken m=n as simpler, and also as more probably the fact. 
I have said that the rays become less easily inflected and deflected ; but it is plain 
MDCCCL. 2 I 
