88 Mr. R. Potter on a particular Modification 



we shall have for the equations of these circles whose centres 

 are in a! and b\ as follows: 



(x-maf + (y-af = i n x* + {y + af = r\ 



and r = r 1 + m a 

 for the central points of interference. 



Substituting, developing, and subtracting, we find 



2 y + mx = mr = m \/ x 2 -\- (y -\- a) z ; 

 raising to the square, and bringing all the terms to one side, 

 we have 



(4- — m 2 ) y 2 + <hmxy—2 m 2 a y — m 1 ^ — 0, 

 which we find to be the equation of an hyperbola. Diffe- 

 rentiating this equation, we find the differential coefficient 

 dy kmy 



dx " 2(4 — m 2 )y-\-4>mx — "27nra 9 



we see that this equation becomes zero when y = 0, but on 

 account of the constant quantity in the equation of the curve, 

 this can only take place at the same time that x is infinite, to 

 fulfil the conditions : hence the axis of the abscissas is tangent 

 to the curve at an infinite distance, and one of its asymp- 

 totes, as we may also learn from the geometrical construction, 

 fig. 5. 



We learn from this, that the central band produced by the 

 interference of two luminous pencils after passing through a 

 prism of glass, should, according to the undulatory theory, 

 nearly coincide with the intermedial line m'n' (fig. 1. and 5), 

 and slightly tend towards the angle of the prism instead of 

 from it, as we find by experiment. Hence the undulatory 

 theory gives no account of this phenomenon. 



According to the other theory, — that light travels through 

 bodies with a velocity which is directly as their refractive in- 

 dices, — on recurring to the general equation 



dr — (k tan $' + h l tan \J/) (poo — j, 



we have, by putting for m its value p, 



d i» = (7i tan <p' + h' tan M) 



(p- — \ = (h tan <p' + V tan J/) {~^~\ 



On referring to fig. 3, we see that we may write for h its 

 identical expression r'dd ; then considering h' as equal to k 9 

 and noting that r increases as <p and <p' decrease, and that 

 hence d<p' must be taken negative, our equation becomes 



a 2 — 1 



df ^ - ? J d<f>' (tan <p' + tan </) ~ ; 



