250 
LORD BROUGHAM’S EXPERIMENTS AND OBSERVATIONS 
flexion, so that A inflects and B deflects through AC, and A deflects and B inflects 
through C B. Let C P=j;, PM= 3/. The force y, exerted by the joint action of A 
and B on any ray passing between them at P, is equal to supposing 
deflexion and inflexion to follow different laws. To find the minimum value of y, take 
its differential dy=0 ; therefore we have 
— x-\-n{a — x)~''~^dx=0, or m{a — xY^^=n{a-\-x)""'^^. 
If m=n (as there is every reason for supposing), then a — x=za-\-x, or x=.0 ■, and 
therefore, whatever be the value of m (that is whatever be the law of the force), the 
minimum value of y is at the point C where A’s deflexion begins. The curve S S', 
which is the locus of M, comes nearest the axis at C, and recedes from that axis con- 
stantly between C and B. Hence it is plain that the fringes must increase (they 
being in proportion to the united action of A and B) from C to B ; and in like manner 
must those made by B’s deflexion and A’s inflexion increase constantly from C to A ; 
and this is true whatever be the law of the bending force, provided it is in some in- 
verse ratio to the distance. 
Proposition X. 
It is proved by experiment that the fringes or images increase as the distance in- 
creases from the direct rays. 
Exp. 1. Repeated observations and measurements satisfy us of this fact. We may 
either receive the images on a chart at various distances from the double edge in- 
strument, approaching the edges until the fringes appear, or we may receive them 
on a plate of ground glass held between the sun and the eye. We may thus measure 
them with a micrometer ; but no such nicety is required, because their increase in 
breadth is manifest. The only doubt is with respect to their relative breadth when 
the edges are not very near and just when they begin to form fringes. Sometimes it 
should seem that these very narrow fringes decrease instead of increasing. However, 
it is not probable that this should be found true, at least when care is taken to place 
the two edges exactly opposite each other ; because if it were true that at this greater 
distance of A from B (fig. 17) they decreased, then there must be a minimum value 
of P M between C and B, and between C and A ; and consequently the law of flexion 
must vary in the different distances of A and B from the rays P, a supposition at 
variance it should seem with the law of continuity. 
Exp. 2. The truth of this proposition is rendered more apparent by exposing the 
two edges to the rays forming the prismatic spectrum. The increase is thus rendered 
manifest. If the fringes are received on a ground glass plate, you can perceive 
twelve or thirteen on each side of the image by the direct rays. It is also worth 
while to make similar observations on artificial lights, and on the moon’s light. The 
proposition receives additional support from these. But care must always be taken 
in such observations, which require the eye to be placed near the edges, that we are 
