346 



p. r: 5 , S 



!• (18) 



ia\=u.,fU. = u\, *^^ = S(.^>7, (20) 



and * , A, are the same functions as before ofu , , ,,u . 



r n,r y n 



A remarkable conclusion may now be drawn from these 

 expressions, by supposing that all the quantities of the 

 form s ^ 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^ ±: Jcx) and sin (ts •±. kx) 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 



ds 

 t L ±: X is large, and that the denominator P — F^ 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\k — k^), making k=: k^ in the numerator, except under 

 the periodical signs, and integrating relatively to k between 

 any two limits which include k", for example between 

 — 00 and -f oo » And because 



S.,v^a\ a\ , zz 1, or rz 0, 



(/^)l h,r h,l ' ' 



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



p, =:a\ , sin^^?,, Q, = a\ , cos ts,, 



t h,\ V t h,l 1 



L =1 Fa' sin (^ts^ — kx^y m^ ~ AV^^ j cos (/.<?^ — kx") 

 and 



