226 
MR. S. S. HOUGH OH THE APPLICATION OF HARMONIC 
7 9 ~ 
a 
r — 2 
(2r - 3)(2r - 1) 
- aL, + 
C 
' +2 
(2r + 3) (2r + 5) 
1 
= 0 , ^ 
• • 
• • 
c„, 
(2?i - 3) (2?i - 1) 
- 
= 0 
From these we obtain 
1 
a_o/a _ T _ (2r + 3) (2r + 5) 
-3)(2r-l) '• CV/C, + 2 
(2r 
_ j __ (2r+ l)(2r +3)^(2r + 5) 
a/C, +2 
and therefore by successive applications 
a_ 2 /C, = (2r - 3) (2r — 1) L, — 
(2r -t- 1) (2r + 3) 
(2r + 1) (2r + (2r + 5) 
L, + 2 — 
1 “j 
(2n - 3)(2n - 1)^(2m + 1) 
(27). 
Tlius the eliminant of equations (27) can be exj)ressed in the form 
111 1 
L c,/c. 
7.9_ 5.7^.9 9.11L13 
■i 
(2w - 3)(2?i - 1)^271 + 1) 
L, 
L, - . . . - 
a 
We therefore see that the roots of the equation A„ = 0 are the roots of 
1 
L., 
5.7C9 9.1P.13 
h - h - . . . - 
(271 - 3)(2n - 1)^2/^ + 1) 
= 0 . 
This form for the equation = 0 has an advantage over the determinantal form, in 
that it enables us at once to proceed to the limit when n is made infinitely great, 
and thus to express the period-equation for the free oscillations by means of the 
transcendental equation 
“2^ ,... = 0.(2S). 
L.J Lg — , , . ad inf. 
