452 Mr. Gr. A. Schott on Frequencies of Free Vibrations 
where as usual real parts alone are to be taken, and 
•=»— 1 r sin 2 (km/n) sin 2 (kiri/n) _ 1 "l 
L =2 \4 sin 3 (wi/n) £sin (wi/n) 2 sin (**'/*) J ' 
»=1 
i=n— 1 
M=^ 
sin (27ffiri/ri) . cos [iri/n) 
8 sin 2 (7ri/») 
'J*.. 
- sin a (iri/n) 4 sin (iri/i 
_ __ f sin 2 (Jari/n) __ sin 2 (kni/n) \ 
1 2 sin 3 (iri/n) 4 sin (iri/n) J * 
Introducing another constant H, where 
tt!T4~ sin2 (**Vtt) 
- tL — ^ 4sin(xi/7i)' 
•=] 
we may also write 
L=J+H-2K, N=2J-H. 
With the same notation as before we have 
H =0, 
TT 1 ^ 
H^JootH-, 
In 
M =0, 
M 1= K-icot^, 
* In 
L„=- 
2K, 
Lj^icot^— — K, 
N„=0, 
N 1 = 2K-J-cot^-. 
Zn 
IX IX ~~~ 1 
H, L, N have their greatest values for k= ^, or — — — ; 
for| moderately large values of n they are respectively -J-K, 
0-017. n 3 , 0-034. n 8 . 
M has a minimum for k= n or - -. 
2' 2 
The constants H, J, L, N are even functions of K, while M 
is an odd function. 
We note the useful series 
H=2 ic ot^i)-, 
s =o 4 2n 
s=k-l 
s=0 2n 
J -=FK-2 A(*-,)(ife-a-l)oot(2!±yi 
= o 2/i 
