202 
MR. L. IvR G. FILON ON THE ELASTIC EQUILIBRIUM OF 
§ 20. Calculation of the Series in the preceding Seetion. 
If we work out the values of fj, gf IJ we find they are as tabulated below : 
/«'• 
yC 
c. 
1 
+-012367 
- -540511 
- -112667 
2 
+-000573 
- -024011 
+-006901 
3 
+ -000078 
- -002538 
+ - 002580 
4 
+ -000025 
- -001259 
+ -000896 
5 
+ -000009 
- -000217 
+ -000381 
6 
+ -000005 
- -000083 
+ -000195 
Hence the parts of the N which depend upon I,!fd, fj/n^, g„'/n~ converge quite 
rapidly enough to allow us to stop after the sixth term. It therefore merely remains 
to evaluate the series 
(— 1)" nirz 
—^ cos - 
c 
and 
A ( — 1)” nrrz 
A . COS — 
1 c 
These cannot be expressed in finite terms, and although we may apj^ly the Euler- 
Maclaurin sum-foi'inula to these series directly, though in a slightly modified form, 
this sum-formula is not really of very great advantage, as its rapidity of convergence 
depends on s, and is such, for certain values of this variable, as to render the formula 
useless as an approximation to the remainder. As a matter of fact, however, the 
series were to Ije calculated only for values of s = z’c/G, i being any integer from 
0 to 6. But foi' such values of z, the cosine terms repeat themselves after n = 6. 
Thus, 
“(-IV'-i rnirt 
cos 
,„3 
= cos V/T/ 6 ( A 
V _ 
1 
ii=o (1 + 127); k 
= 
Z V - 
r‘ 
cos 2«7r/6 ( X 
1 
Vm=o (2 -1- 
=0 (I F i27?iy 
1 
=0 (8 + 12»2f 
+ 
— cos 'ITT 
rii-= CO 
S' - 
i=o (G + 12'rt()'* 
A precisely similar formula holds for 
+(-i) 
1)1=00 
•w=o (12 + 12??i)^/ 
COS 
}i7ri\ 
