

146 EXPERIMENTS AND INVESTIGATIONS 



action , provided it be inversely as some power of the distance, 



produce fringes or images which increase with the distance 



from the direct rays. 



Let (fig. 17) A and B be the two bodies, and AC = C B = a 

 be their spheres of flexion, so that 

 A inflects and B deflects through 

 A C, and A deflects and B inflects 

 through C B. Let C P = ar, P M = y. 

 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 



(a + x} m (a-xY 



to follow different laws. To find the minimum value of y, 

 take its differential dy = ; therefore we have 



m(a -f x)~ m - l dx + n (a x)~ n ~ l dx = 0, or m(a #) B+l 

 = n (a + x) m+l . 



If m n (as there is every reason for supposing), then 

 a x = a -\- x, or a; = ; 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 constantly 

 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 inverse ratio to the distance. 



PROPOSITION X. 



It is proved by experiment that the fringes or images 

 increase as the distance increases from the direct rays. 



Exp. 1. Kepeated observations and measurements satisfy us 

 of this fact. We may either receive the images on a chart at 

 various distances from the double edge instrument, approach- 

 ing the edges until the fringes appear, or we may receive 

 them on a plate of ground glass held between the sun and the 



