DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
169 
oscillations about steady motion of a weight suspended by a string of which the 
other end is made to describe uniformly a horizontal circle, in the case in which the 
string intersects the vertical drawn downwards from the centre of the circle described 
by its upper end. This motion is not, however, secularly stable when there is 
Dissipativity (Thomson and Tait, as above, p. 388); and, of course, not instantaneously 
stable, the roots of the corresponding quartic equation having real parts of which some 
are positive. 
A second remark relates to the generality of the form of the differential equations 
used in the text. Equations such as 
oL/ST\ 
dt\dxj 
ST 
dx r 
■ firiXi + 
8 x + §!_ + ^ = O 
PrnX n -r -V 4r, 
where /3 rs is a function of ..., 
/3 U ..., (3 n in the form 
x n capable of expression in terms of' n functions 
; r ‘ dx s dx r 
are included in this form, with a slight modification due to the presence of the 
Dissipativity F, and the supposed non-conservative forces Q r . For this it is only 
necessary to take 
L - 1 + + . • • + fi;,X n 
it • SL • SL 
H = a?! — + ...+x n x-r- 
Sa 1 ! ox n 
-V, 
■ —L, 
and to eliminate x 1} x n , in the familiar way, from the n equations 
SL 
Then the final equations are 
X — 
SH 
°y r ' 
SH SF , n 
Vr dx„ S,A 
Particular illustrations are: (l) the equations of Thomson and Tait (as above), p. 392, 
for which the coefficients (3 rs are constants. Then we may take /3 r = c rl x x -{-... +c rn x n , 
where the constant coefficients c rs are - in part arbitrary; (2) the equations of Lord 
Kelvin for liquid motions of ring-shaped solids, ‘ Collected Papers,’ IY. (1910), p. 106 ; 
(3) the equations of motion of a system relatively to a rotating frame (Lamb, ‘ Hydro- 
dynamics,’ third edition (1906), p. 294. Cf. Thomson and Tait,' as above, § 319, p. 307, 
and p. 319), for which we may take, if (f, y) be the co-ordinates of a point of the 
system relatively to the rotating frame, 
VOL. CCXVI.— A. 
