114 MR. L. N. G. FILOX OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
value of the corresponding term in the remainder is seen to become very small of the 
order ijr when r is made very large, n remaining finite. The value of the whole 
remainder is therefore also small of the order Ijr. Consequently this remainder 
tends to zero as we make r large, and the series is therefore a true arithmetical 
equivalent of J„. 
We have still to show that a similar result holds for the expansion found for I, 
namely, that the integral we have called Il„ tends to the limit zero when n is 
indefinitely increased. This we can do as follows :—■ 
IC = (-1)" 
is numerically less than 
siiih rtj + cosh u , 
~ cos uz du 
Jo sinh” 2?^ sinh 2zt + 2u 
(2?d'‘ siuh u + 'll. cosh u 
sinh" '2u sinh 2u + 2u. 
da. 
and it is easy to show that both {2«.)Vsinh" 2u and (sinh u + a coshy6)/(sinh 2u + 2u) 
continually decrease as u increases. 
Hence, if we split up j into + j , the first part is less tnan [oj X value of the 
0 ♦ 0 *-'10 
integrand vdien u — 0], i.e., <i a>/2. The second part is also less than 
{2(oY 
sinh u + u cosh u 
(sinh" 2m) Jo, sinh 2u + 2u 
du. 
Denoting the last integral, which is finite, by M, we have < y + 
numerically. 
But 
(2m)" 
sinh" 2 
M 
< 
(2m)" 
A + i 
< 
mA" 
< 
1 , ' 
1 + 
9, 
nco~ 
Therefore < y + 
:m 
1 + 
2vor 
Now if M be chosen equal to rr\ R,, < i quantity which tends to 
zero when n tends to infinity. R,, itself therefore tends to zero for all values of 2 , so 
that the series (93) may be extended to infinity. 
§ 26. Deductions as to the Rapidity with tvhich the Local Disturhanees die out as we 
leave the neighbourhood of the Load. 
If we look at (93) and perform the differentiations, then, remembering that f) 
is of degree (2^ — 1) in 2 , ( 2 ) and y^ioj ( 2 ) are of degree 2t in 2 , and ^21 + 1 f) is of 
degree (2( + 1) i^^ only terms occurring in I will be of the form (algebraic 
polynomial in 2 ) X (sech y- or sech-y , or their differential coefficients j. Now 
