140 REPRODUCTION OF EULER’S MEMOIR OF 1758 
so that the value of the original radical is 
v {(-)} = cos U. 
v (O') 
Substituting in the expressions for the cosines of the arcs l, m, n, these values of v 
and the radical; the formulae after some reductions become 
cos 
2)Ap BCp (m - №3) V (G- 2) 2 ) • J T LAqr V (G - D 2 ) r T 
~ ~G + Gy/(K-2LMNGuj S u + V (0) V (K - 2LMNGu) C0S u ’ 
2)Po (Llo (OT - L(S) V (0 - 2) 2 ) . r7 , MBrp V (0 - 2) 2 ) , TT 
cos m £ + g*J(K-2LMNGu) SmU+ ^ (G) *J(K- 2LMNGu) °° ’ 
2)Cr ABr(Z8-Jim)V(G-2> a ) • NC V qs/(G- 2) 2 ) 7r 
cos w - g, + G,j(K- 2LMNGu) sm U + (G) (K - 2LMNGu) cos ’ 
where for shortness p, <7, r are retained in place of their values V (2Zw + 21), \/ (2Mu + 33), 
V(2 JVit + (5). 
The values of Z, m, n being known, that of \ could be determined by the 
differentia] equation 
7 dt (q cos m + z cos n) 
d\ = — ,, 7 , 
sm 2 Z 
and then the values of p, v would be determined without any further integration; 
but it is better to consider, in the place of any one of the principal axes in particular, 
the instantaneous axis, which is a line inclined to these at angles a, /3, y, the cosines of 
which are — - (if as before to 2 = p 2 + q 2 + r 2 ). Considering the instantaneous axis 
(0(0(0 
as a point of the sphere, let j denote the distance OP from the fixed point P, and 
(f> the inclination OPQ of this distance to the fixed arc PQ. We have 
cos j = cos a cos Z + cos /3 cos m + cos 7 cos n, 
sin j cos (f> = cos a sin Z cos \ + cos /3 sin m cos p + cos 7 sin n cos v, 
sin J sin (f> = cos a sin Z sin \ + cos /3 sin m sin p + cos 7 sin n sin v,