136 
REPRODUCTION OE EULER’S MEMOIR OF 1758 
[404 
and introducing the auxiliary quantity u such that du = pqrdt, we have 
p- = $[ + 2 Lu, 
q 2 = 33 + 2 Mu, 
r' 2 ~(S +2 Nu, 
where 51, 53, 6' are constants of integration, and thence 
f du 
J V {(81 + 2Lu) (53 + 2Mu)) ((5 + 2Nu) ’ 
where the integral may without loss of generality be taken from u = 0; u, and 
consequently p, q, r, are thus given functions of t\ and it is moreover clear that 
Si, S3, (£ are the initial values of p 2 , q 2 , r 2 . We have also if <y be the angular 
velocity round the instantaneous axis 
or = 21 + 23 + £+ 2 (L + M + N)u. 
Euler then assumes that the position in space of the principal axes is geometrically 
determined as follows, viz. (treating the axes as points on a sphere) it is assumed 
that the distances from a fixed point P of the sphere are respectively l, in, n, and that 
D 
the inclinations of these distances to a fixed arc PQ are respectively X, p, v. We have 
then the geometrical relations 
sin (p, — v ) = — 
cos 2 1 + cos 2 m + cos 2 n = 1 ; 
cos l 
whence also 
, x cos m cos ii 
— •— , cos (/X — v ) = ; r-—, 
sm m sin n ' sin m sin n 
. , % x cos m / . . cos n cos l 
sin h - A) = — .—j , cos (v —\) = ^ —- , 
sin n sin l / sin n sm l 
• /. x cos ii cos l cos m 
sin (X - p) = . . , cos (X - p,) = . ; 
sin l sin m n sin l sm m 
sin /x = 
cos p, = 
sin v = 
cos v = 
— cos X cos 11 — sin X cos l cos 111 
sin l sin m 
sin X cos n — cos X cos l cos 111 
sin l sin m 
cos X cos m + sin X cos l cos n 
sin l sin n 
— sin X cos m — cos X cos l cos n 
sin l sin 11