84 Mr. A. Schuster on the Elementary Treatment 



the subsequent maxima and minima agreeing still more closely 

 with the first approximation. 



The elementary method which has been described gives 

 therefore results which for an approximate method are of 

 surprising accuracy, if it is considered that no integration 

 has been made use of beyond that implied in the statement 

 that if a large number n of vibrations of equal amplitudes 

 and phases uniformly distributed through two right angles are 

 added up together, the amplitude of the resultant is 2??a/7r. 



The more general proposition, that if a number of vibra- 

 tions of equal amplitude have phases uniformly distributed 

 through an angle 20, the resultant is obtained by reducing 

 in the ratio of sin 6 : 6 the value calculated on the supposition 

 that they have all the same phase, has many important applica- 

 tions in the theory of diffraction. Several pages are generally 

 devoted to the calculation of the position and intensity of 

 diffraction-fringes produced by a slit on a screen at a great 

 distance. The results obtained may be written down at once 

 from the above proposition, which may be deduced with the 

 help of a series; so that a satisfactory account of the position 

 and intensity of diffraction-bands can be obtained without the 

 help of the integral calculus. 



The series 



Seee???,— m 2 + m 3 — m A + . . . +m- n , . . . . (1) 



where the terms alter their value only slowly, occurs frequently 

 in problems on diffraction, and its sum is said to be equal to 

 half the first added to half the last term ; but I have not met 

 with a satisfactory proof of this statement. When a proof is 

 attempted the second term is balanced against half the first, 

 and half the third, and so on, but no reason is given why the 

 balance should not be made in another way, and the second 

 term, for instance, balanced against three quarters of the first 

 and one quarter of the third term. The following considera- 

 tions will show in what cases the addition of the above series 

 may be effected in the manner indicated. We may write the 

 series 

 m. , /m 1 , m„\ , /ra 3 m 5 \ , 



s s i + (t - m2+ 2 r (t ~ mi + ir ■ ■ • 



+ (^ + *.-!+ f)+^ • -(2) 



T~ L("2-~" !3+ 27 + U ' "" + t) + --- 

 *(— 2 ^n-2+-y-JJ 2 _ +m. . .(3) 



or 



