478 Transactions of the Society. 



which gives the resultant amplitude at any point u as a function 

 of u and a. If n > a, we have 



A , \ f + x sin u , , C u 



A (u) = du -f 



J -co u J 



- a sin u 7 

 — — du 

 u 



■r- 



sin w 



w 



rf« (80) 



By (78) the first term is equal to ir. 



The integral in (79), (80) is known as the sine-integral. In 

 the usual notation 



s: 



l JL^ du = si (x) (81) 



o U 



so that (79) may be written 



A (u) = 7r — si (a — u) — si (a + u) . . (82) 



and (80) may be written 



A (u) = ir + si (u — a) — si (u + a). . (83) 



The function si has been tabulated by Dr. Glaisher.* 



At the centre of the geometrical image of the bar, u = 0, and 

 (82) becomes 



A (0) = 7T - 2 si (a) (84) 



If x is small, (81) gives 



'''' — rro + 5.i.i!u5 -" i (85) 



so that in (82) if a be small, 



a / \ o ,2a (a 2 + 3 u 2 ) /OPN 



A (u) = 7T - 2 a + — 3 V ^ ^ - (86) 



From this we see that over the whole geometrical image of the 

 bar the amplitude of vibration is nearly the same. If we write 

 I for the intensity, where I (w) = {A (tc)\ 2 , and denote by I the 

 value of I corresponding to a uniform ground (a = 0), then 



L - I 4a 



(87) 



1 7T 



This gives the proportional loss of illumination over the image 

 of the bar, and it suffices for the information required near the 

 limit of visibility. For example, if the loss of light over the 



* Phil. Trans., 1870. 



