ON THE SPECIAL PROBLEMS OF DYNAMICS. 231 
dw dv 
dm{ y ——2z— }=0 
z (y ae edt i 
a du o)=°, 
iy Fn at 
dv dy\ _ 
Sdm (« aF —vf) =0. 
It is assumed that at any moment the body revolves round an instantaneous 
axis, inclinations a, 3, y, with an angular velocity s ; this gives 
u=b(zcos B—ycos y) = gz —ry, 
U=8(v COS y—Z COS a) = 1X —pz2, 
w=s(y¥ cos a—w cosh) =px—qy, 
if 8 cos a, 8 cos 3, 8 cos y are denoted by p, g, 7. The values of du, dv, 
dw are obtained by differentiating these formule, treating p, 7, 7, %, y, 2 as 
yariable, and replacing dx, dy, dz by udt, vdt, wdt respectively; in the 
resulting formule for ydw—zdv, &c., w, y, ¢ are considered as denoting the 
coordinates of the element in regard to axes fixed in the body and moveable 
with it, but which at the moment under consideration coincide in position 
with the axes fixed in space. The expressions for 3 (ydw—zdv) dm involve 
the integrals A= ef" (y? +2") dm, &c., where the coordinates refer to axes fixed 
in the body; and if these are taken to be principal axes, the expression for 
3 (ydw—zdv) dm is =Adp+(C—B) qrdt, which gives the three equations 
Adp+ (C—B) qrdit=0, 
Bdq + (A—C) rpdt=0, 
Cdr + (B—A) pqdt=0, 
already referred to as Euler’s equations. 
170. Next, as regards the determination of the position in space of the 
principal axes: if about the fixed point we describe a sphere meeting the 
principal axes in w,, y,, z,, and if P be an arbitrary point on the sphere and 
PQ an arbitrary direction through P, the quantities used to determine the 
positions of x,, y,, 2, are the distances w,P, y,P, z,P (=1, m, ”) and the incli- 
nations «,PQ, 7,PQ, z,PQ (=A, p, v) of these ares to the fixed direction PQ 
(it is to be observed that the sines and cosines of the differences of A, p, v are 
given functions of the sines and cosines of /, m, n, and, moreover, that 
cos*/-+ cos’m+cos*x=1, so that the number of independent parameters is 
three). The above is Euler’s definition ; but if we consider a set of axes fixed 
in space meeting the sphere in the points X, Y, Z, then if the point X be 
taken for P, and the arc XY for PQ, it is at once seen that the angles used 
for determining the relative positions of the two sets of axes are the same as 
in Euler’s memoir “ Formule Generales, &c.,” 1775 (ante, No. 132), where 
the formule for this transformation of coordinates are considered apart from 
the dynamical theory. 
Euler expresses the quantities p, g, 7 in terms of an auxiliary variable u, 
which is such that du=pqrdt; p,q, r are at once obtained in terms of u, 
and then ¢ is given in terms of w by a quadrature. Euler employs also an 
auxiliary angle U, given in terms of u by a quadrature. And he obtains 
finite algebraical expressions in u, cos U, sin U for the cosines or sines of 
l,m,n; s(the angular distance IP, if I denote the point in which the instan- 
taneous axis meets the sphere), ¢ (the angle IPQ) and A—9, p—9g, v—¢. 
