THE HON. J. AY. STKUTT ON THE THEOKY OF EESONANCE. 
85 
An application of Lagrange’s method gives as the differential equations to the motion, 
x, 2 X, — X 2 
-+« — s— 
= 0, 
■^2 I ^2(^2 X, Xg X 3 ) j. 
c 2 +a \ S + S' /- U ’ 
(10) 
v \ r _ y 
±* + a*h*±* = Q. 
By addition and integration 
Y X Y 
- 1 + - 2 +- 3 =0. 
c l C 2 C 3 
Hence, on elimination of X 2 , 
x > + s + ^)x, + if X3 } - ( | 
X 3 +^(c 3 +<>X 3 +^XA = 0 
'S' 
Assuming 'X 1 =Ae pt , X 3 =Bs p< , we obtain, on substitution and elimination of A:B, 
])* +pW | J -y— - + -^7 ~ j + ss' + c 2 (c 1 -f-c 3 )j =0 (11) 
as the equation to determine the resonant notes. If n be the number of vibrations per 
p- 
second, n-— — the values of jT given by (11) being of course both real and negative. 
The formula simplifies considerably if c 3 =c„ S'=S; but it will be more instructive to 
work this case from the beginning;. Let c, =c., — nic.,=mc. 
The differential equations take the form 
X *+§{( 1 + W ) X ^ X 3j =0 ’ 
X,+${(l+m)X 3 +X 1 'l=0, 
Hence 
while X.,= 
_x i± x 3 
m 
(Xi+XJ'fi- g (ot+2)(X,+X 3 )=0, 
(Xj— X 3 )"-j-^m(X 1 — X 3 )=0. 
The whole motion may be regarded as made up 
of two parts, for the first of which X 1 +X 3 =U ; which 
requires X 2 =0. This motion is therefore the same 
as might take place were the communication between 
S and S' cut off, and has its period given by 
Fig. 
„ (re, a*mc 
N 
MDCCCLXXI. 
