NEWTON'S EXPERIMENTS ON DIFFRACTION. 103 



the front of the wave where it enters the external aperture to be divided 

 into a great number of small parts ^v ; and suppose each of these to 

 be the origin of a small wave which diverges from it as a center. The 

 distance from the point v of the aperture to the point w of the slit is 



^{a' + {v-wY]=a+ —(v-wy; 



and the disturbance produced at w by the small wave spreading from the 

 space Sv at v will therefore be proportional to 



^tj.sin. — {vt- A — a— —-(v — wY]. 



Integrating this with respect to v, the coefficient of sin — {vt — A- a) 

 will be 



L cos ~{v-wy, 



and the coefficient of cos —- (\t— A — a) will be 



A 



-Xsin^(«-M;)^ 

 The first of these integrals = X cos ^ iv \/ —r — w V -r-J '' 



and putting ^(s) for f. cos f- »M, this integral between the limits v = a, 

 v = l3, will be proportional to 



<h\^\/ -- —W\/ -—] — d>\a \/ -— - W\/ ^\. 



(TV \ TT 



- xM , the integral - /„ sin -— (v — ivy 



between the same limits will be proportional to 



- ^ l*^ ^ - ■" ^^) + H° ^Fx - =" ^i)' 



o 2 



