346 
X n =—I. 
P= 8, 21 (4,8, sinds,. = at An Anis | 
(18) 
Ee 2% ‘ 
@, = 2,1 (A;,,. 008 ds, « Lay Air a1)? 
na 
= Qi Xe, i (19) 
eS PS a Ee oy agai 
ka’ =u,; ka =u, — ae? (20) 
and S54, are the same functions as before of Uys wu. 
’ 
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 ¢s, are periodic. For in this case the func- 
tions cos (¢s_ + kw) and sin (¢s, = kx) will vary rapidly, and 
pass often through all their fluctuations of value, between 
the limits 1 and —1, while & and the other functions of 
that variable remain almost unchanged, provided that 
ut + wx is large, and that the denominator /* — &” is not 
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 — i), making k= in the numerator, except under 
the periodical signs, and integrating relatively to £ between 
any two limits which include /’, for example between 
—o and +o. And because 
mes ‘ = es 
Aa kl 1, or =0, 
according as r = 1 or >1, we may make 
‘4 hed — A 
E. =Any sin tS a= Any cos ts, 
