by a Screen bounded by a Straight Edge. 421 



It will be assumed that the wave-fronts are parallel to the 

 diffracting edge and that the disturbance has lasted only for 

 a finite time. We take rectangular axes, the diffracting 

 edge being the axis of £ and the axis of y perpendicular to 

 the wave-fronts, the waves approaching along the negative 

 half of this axis. The distance from the diffracting edge 

 will be denoted by tsr, t will be written for the time, and 

 c for the speed of propagation of the disturbance. 



V, the measure of disturbance, must satisfy the differential 

 equation 



[v-|g]v=o (i) 



The incident wave is determined by an equation such as 



Y = f(ct- 1/ ), (2) 



where ty is any continuous function satisfying the condition 

 that, for all values of T greater than some constant K, 

 i/r( — cT) and its derivatives are zero. 



It is proposed to construct an expression suitable for 

 representing the disturbance diffracted into a shadow. For 

 this purpose consider tl.e integral 



/= 



/"»00 



\ f(ct-y-u%)Udu, .... (3) 



in which £ represents ot — y and U is a function of u. 



The parameters of yjr in this integral range from ct — tu 

 to — so and suggest the passage of elementary waves by the 

 direct line from the edge to x, ?/, z and by longer routes. 

 If the integral / can be made to satisfy the fundamental 

 differential equation (1) it may serve for constructing the 

 solution of the diffraction problem. 



It is easy to verify that 



differentiation of yjr being represented by dashe?. 



The integration can be effected if the expression in the 

 larger brackets is a perfect differential. The condition for 

 this, viz., 



|L{2(«S-u)U}=«U, (5) 



is satisfied if 



2(m 2 -m)U = C(m-1)^ (6) 



