346 



(18) 



Q. = S, * (a, cos ts . 2_ " A, a\ , V 

 / 0)1 v h,r r (h)l h,r h,!/' J 



x-^'\a\x ., (19) 



ka\ ="«; , AV. = «\ , F 2 = S, -" a\ 2 , (20) 



and * , a, are the same functions as before of w, .... « . 



»• A,r 1' n 



A remarkable conclusion may now be drawn from these 

 expressions, by supposing that all the quantities of the 

 form s 2 are not only real but positive, so that the functions 



cos ts and sin ts are periodic. For in this case the func- 

 tions cos (ts ± kx) and sin (is ± lex) will vary rapidly, and 

 pass often through all their fluctuations of value, between 

 the limits 1 and —1, while k and the other functions of 

 that variable remain almost unchanged, provided that 



t 21 d= x is large, and that the denominator k 2 — A-' 2 is not 



dk 

 extremely small. We may therefore in general confine 



ourselves to the consideration of small values of this deno- 

 minator; and consequently may put it under the form 

 2k y (k — 1c), making k s= Jc in the numerator, except under 

 the periodical signs, and integrating relatively to k between 

 any two limits which include k\ for example between 

 — oo and -f- oo . And because 



2 'V, a\ . = 1, or = 0, 



001 h,r h,l ' 



according as r zz 1 or > 1, we may make 



p, — a\ . sin ts. , Q t zz a\ , cos ts., 



t li,\ 1 t h,\ 1 



l = li a s sin (ts x — kx), M^ — k K A' h x cos (ts — kx) 

 and 



