37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 
243 
posed to become variable. We have, in this case, by Lagrange’s theory of the variation 
of the arbitrary constants, the formulae 
where 
and in which V is supposed to be expressed as a function of a, b, c, d, e, f, t. 
Thus the solution of the problem requires the calculation of thirty coefficients (a, b), 
or rather of fifteen only, since evidently (a, b) = - (b, a). It is known that these coeffi 
cients are functions of a, b, c, d, e, f without t ; so that, in calculating them, any assumed 
arbitrary value, e.g. ¿ = 0, may be given to the time. 
In practice, it often happens that one of the arbitrary constants, e.g. a, may be 
expressed in the form 
a = F (X, fi, v, u, v, w, t, b, c, d, e, /), 
where b, c, d, e, f are given functions of X, /i, v, u, v, w, t. In this case, it is easily 
seen that we may write 
(a, b) = {(a, b)} + (c, b)^ + (d, b)^ + (e, b) g + (/. b) 
da 
df’ 
where, in the calculation of {{a, b)), the differentiations upon a are performed without 
taking into account the variability of b, c 
In the particular problem in question, the following are the values of the new 
variables u, v, w {Math. Journal, memoir already quoted, [6]), 
(29), 
2 
v = K ( vAp + Bq — XCV), 
equations which may also be expressed in the form 
QAp — ( 1 -f- X“) u + (Xyu. 1>) v -f- {v\ — yii) w 
ZBq = (X/a — v) u + ( 1 + ¡j?) v + {pv + X) w, 
2Cr = (v\ + /jl) u + {fir — X) v + ( 1 + v 2 ) w, 
(30), 
or putting for shortness 
\u + fiv + Viv = «T 
....(31), 
31—2