170 
PROF. H. F. BAKER ON CERTAIN LINEAR 
The equation of energy in general is at once seen to be 
—+20x 
dt dt r dx r ^ rr ’ 
so that if H be explicitly independent of the time, the forces Q r be absent, and F be 
a homogeneous quadratic function of ,x l5 x„, 
dH 
dt 
= —2F.] 
§ 17. The simplicity of the formulation depends on the fact that the invariant 
factors of /3 _1 a — \fs are linear. We have obtained this by assuming that the form ap 2 
only vanishes when every element of p is zero. But the invariant factors may be 
linear when this is not so, and the roots of the determinantal equation are not pure 
imaginaries. For instance, take Hill’s equations for the motion of the moon, under 
certain limitations, 
'' i / ^ _o ^ y I I ny 
dt ,^nj t+[ y,np = 0, |f + 2„ i + ^ = 0. 
W riting 
these are the same as 
F = — — — fn 2 £c 2 +|- (X+ra/) 2 +|- (Y — nx) 2 , 
dx _ 0F dX _ 9F dy _ 3F dY _ _ 3F 
dt 3X ’ dt dx ’ dt 0Y ’ dt dy' 
The so-called moon of no quadratures is obtained by variation from the solution 
expressed by 
x — cr, X = 0, y — 0, Y — no-, 
where a- is given by fx = 3wV 3 ; this is a position of relative equilibrium. The 
matrix S- of the notation used above is zero ; the matrix a is 
j —8 n 2 
0 
" y -- - 
0 
—n \ 
( 0 
1 
n 
0 1 
0 
n 
4 n 2 
0 1 
\ 
\ —n 
0 
0 
1 / 
In this case the quadratic form up 2 
is 
- 9 n% 2 + (p 2 + 2npz) 2 + {np 1 -p.tf 
