404] 
ON THE ROTATION OF A SOLID BODY. 
137 
llie geometiical equations connecting the resolved angular velocities p, q, r with 
the differentials of l, m, n, \ p, v are 
dl sin l — dt(q cos n — r cos to), d\ sin 2 1 = — dt(q cos to + r cos n ), 
dm sin m dt(rcosl — pcosn), d/i sin 2 to = — dt (r cos n -{-pcosl ), 
dn sin n — dt(p cos to — q cos l ), dv sin 2 n = — dt (p cos l + q cos to). 
Multiplying the equations of motion respectively by cos l, cos to, cos n, and adding, 
we obtain an equation which is reducible to the form 
d {Ap cos 1 + Bq cos m -f Gr cos n) = 0, 
Ap cos 1+ Bq cos m + Gr cos n — 2), 
whence integratin' 
2) being a constant of integration. One other integral equation is necessary for the 
determination of the angles l, m, n. The expressions for dl, dm, dn give at once 
p dl sin l + q dm sin m + r dn sin n = 0. 
Instead of the arcs l, m, n, Euler introduces a new variable v, such that 
v = p cos l + q cos m + r cos n; 
by means of the last preceding equation, we find 
dv = dp cos l + dq cos m + dr cos n, 
and then, substituting for dp, dq, dr, their values, 
, " ’ ilfcosra 
dv= + + 
d 
from which the relation between v and u is to be determined. We have 
cos 2 l + cos 2 m + cos 2 n — 1, 
Ap cos l + Bq cos m + Gr cos n = 2), 
p cos l + q cos m + r cos n = v, 
which give cos l, cos m, cos n in terms of u, v; the resulting formulas contain the 
radical 
/ ((L-Adfr- + M-B-r-p 2 + N-G'Yf) - 2) 2 (.x 2 + y 2 + z 1 ) ) 
Y 1 + 2Dv (Ap* + Bq 2 + Gr 2 ) - v 2 (A 2 p 2 + B 2 q 2 + GV)j ’ 
which for shortness is represented by V{(")}* W e then have 
Dp (NCq 2 - MBr 2 ) + BGpv (Mr 2 - JS T g 2 ) + LAqr V{(~)} 
cos l - L 2 A 2 q 2 r 2 + M 2 B 2 r 2 p 2 + N 2 C 2 p 2 q 2 
<S)q (LAr 2 - NGp 2 ) + GAqv (Np 2 - Lr 2 ) + MBrp V{(-)} 
cos m ~ UA 2 q 2 r 2 + M 2 B 2 r 2 p 2 + N 2 Cy<:f ’ 
Dr (MBp- - LAq 2 ) + ABrv (.Lq 3 - Mp 2 ) + NGpq V{(-)} 
cos n - pA 2 q 2 r 2 + M 2 B 2 r 2 p 2 + N 2 C 2 p‘ 2 q 2 ~ 
C. VI. 
18