682 Lord Rayleigh on Hamilton's Principle and 
from which it appears that U is in reality a function o£ x\ y\ 
Z, m. As equivalent to (13), we have 
x^dJJ/dl, y = dV/dm, . . . (14) 
r = dXJ/dx', m'=dU/dy'. . . . (15) 
So far U appears as a function of the four variables x', y\ 
Z, m ; but from its nature, as dependent upon la: + my and V, 
and from the axial symmetry, it must be in fact a function of 
the three variables 
x' 2 + y' 2 , l 2 + m 2 , and lx' + my', 
the latter determining the angle between the directions of 
x\ y' and I, m. When these quantities are small, we 
may take 
U = U( >+U( 2 >+IP> + , . . . (16) 
where U (0) is constant and 
U( 2 )=lLG 2 + m 2 ) + Mry/+y'7n) + lN(^' 2 +,v /2 ), . (17) 
L, M, N being constants. If we stop at U (2) , equations (14) 
give 
#=LZ + M#' f y = Lm + My / , . . (18) 
determining #, y as functions of as', y\ /, m. We have next 
to introduce the supposition that x, y is conjugate to x\ y' * 
Hence L = 0, for to this approximation x, y must be deter- 
mined by x', y' independently of /, m. Accordingly, 
x = kx', y = W (19) 
We are now prepared to proceed to the next approximation. 
In order to correspond, as far as may be, with the notation of 
Seidel * we will write 
U< 4 > = lk(l 2 + m 2 ) 2 + B(l 2 + m 2 )(lx' + mij) 
-\-^{C-D){lx / + my / ) 2 + W(l 2 + m 2 ){x' 2 +y' 2 ) 
+ E(lx' + my'Xx /2 +y' 2 )+F(x' 2 +y /2 )\ . . . (20) 
amich is the most general admissible function of the 
fourth degree. 
From (20) we obtain by use of (14) the additional terms 
in x and y dependent on U (4) . No generality is lost if at 
this stage we suppose, for the sake of brevity, y = 0. 
Accordingly, 
x = AZ(/ 2 + m 2 ) + B#'(3Z 2 + m 2 ) + QaJ 2 l+ Ea J \ [ (21) 
y = Am(l 2 + m 2 ) + 2Bx'lm + Vx' 2 m (22) 
* Finsterwalder, Miinclien. Sitz. Ber. xxvii. p. 408 (1897). 
