of Edinburgh, Session 1870-71. 
389 
where I denotes the resultant “ impulse ” of the cyclic motion 
when the solid is at rest; Z, m, n its direction cosines; Gr its 
“rotational moment,” (§ 6, “Vortex Motion”); and «, y, z the co¬ 
ordinates of any point in its “ resultant axis.” These (14) with 
(13) used in (6) give the equations of the solid’s motion, referred 
to fixed rectangular axes. They have the inconvenience of the 
coefficients [w, w], [u, v], &c , being functions of the angular co¬ 
ordinates of the solid. The Eulerian equations (free from this 
inconvenience) are readily found on precisely the same plan as 
that adopted above for the old case of no cyclic motion in the 
fluid. 
The formulas for the case in which the ring is circular, has no 
rotation round its axis, and is not acted on by applied forces, though 
of course easily deduced from the general equations (14), 13), (6), 
are more readily got by direct application of first principles. Let 
P be such a point in the axis of the ring, and (£, A, B, such con¬ 
stants that M^co 3 + A u 2 + Bv 2 ) is the kinetic energy due to 
rotational velocity co round D, any diameter through P, and trans¬ 
lational velocities u along the axis and v perpendicular to it. 
The impulse of this motion, together with the supposed cyclic 
motion, is therefore compounded of 
A u + I along the axis 
Be perpendicular to „ „ , 
momentum in lines through P 
and moment of momentum ffw round the diameter D. 
Hence if OX be the axis of resultant momentum ; (x, y) the 
co-ordinates of P relatively to fixed axes OX, OY; 0 the inclina¬ 
tion of the axis of the ring to 0; and £ the constant value of the 
resultant momentum : we have 
£ cos 6 — A u + 1; — £ sin 0 — Ba , 
£y = ; 
(15.) 
and 
x — u cos 6 - v sin 0 ; y — u sin 0 + v cos 0 ; 9 = w 
3 G 
VOL. VII. 
