168 
PROF. H. F. BAKER ON CERTAIN LINEAR 
take the forms 
dX x dY 1 dX 2 
dt dt dt 
dY;■ 
of 
O+e) (x„ Y„ x 2> x 2 ), 
where <x denotes the matrix above written, with only diagonal elements ia- 1 , &c., 
and 0 is the matrix 
e = h-'p-'bh. 
The solutions of these equations are then expressed by 
(x„ Y„ x 2 , y 2 ) = QO+e) (x,«, YA X 2 “, y 2 "), 
where Xd, Yd, are the initial values. Now, by a previously given formula, 
Q (<r + 0) = Q {a) Q [Q- 1 (a-) 0Q (<r)], 
where il (a) has the simple form 
the solution is thereby expressed in powers of the small quantities occurring in 
The preceding work hafe wide applications; a particular case is that of the 
oscillations of a dynamical system about a state of steady motion, for which S', and 
0, is zero. 
[ October 30 , 1915. —To prevent misunderstanding, two remarks may be added to 
§16. The condition that the quadratic form ax 2 should be positive, though sufficient, 
is not necessary in order that the roots of the determinantal equation (/— ^) = 0 
should be pure imaginaries. For instance, if a, b, u, v be real positive constants, and 
H be a quadratic form 
H = \a (yx-nx,) 2 +%b (y 2 -mx 1 ) 2 - ^ x x 2 -x 2 2 , 
the motion about x 1 = 0, x 2 = 0, y 1 = 0, y 2 = 0 expressed by the equations 
= 3H/ 02 / a , ijx = -9H /dxx, x 2 = 3H /dy 2 , y 2 = — 0H/0x 2 , 
is instantaneously stable if ah {m—n ) 2 > (u+v) 2 , the corresponding quartic equation 
having all its roots purely imaginary. This essentially is the case noticed by 
Thomson and Tait, ‘ Natural Philosophy,’ I., pp. 395, 398, where the illustration is 
that of a gyrostat balanced on gimbals. A simple illustration is also that of the 
