387 
of Edinburgh,. Session 1870-71. 
One chief object of this investigation was to illustrate dynamical 
effects of helipoidal property (that is right or left-handed asymmetry). 
The case of complete isotropy, with heliyoidal quality, is that in 
which the coefficients in the quadratic expression for T fulfil the 
following conditions. 
[w, u] = [v } v\ = \w, w\ (let m be their common value) \ 
[^r, w] = [p, p] = [cr, cr] ,, n „ ,, „ 
K w] = b,p] = O, <r] „ h „ „ „ (10). 
[v, w] = [«;, w] = [w, F] = 0 ; [p, cr] = [cr, zff] = [^r, p] =0 
and [u, p] = [' u , a-] - [v, cr] = [v, w] = K «r] = [w, p ] = 0 J 
so that the formula for T is 
T = ^{m(y + v 2 +.w 2 ) + w(^ 2 + p 2 + o- 2 ) + 2h(u<v + vp-\-'W(r)} . (11) # 
For this case therefore the Eulerian equations (1) become 
d( mU (U “ m ( W — wp)=X, &c. 
and *& + »") = Lj * c . 
dt ’ 
[Memorandum: — Lines of reference fixed relatively to the 
body]. J 
But inasmuch as (11) remains unchanged when the lines of 
reference are altered to any other three lines at right angles to 
one another through P, it is easily shown directly from (6) and 
(9), that ; if, altering the notation, we take u , v, w to denote the 
components of the velocity of P parallel to three fixed rectangular 
lines, and w, p, cr the components of the body’s angular velocity 
round these lines, we have 
d(mu + h<zr) _ \ 
dt ’ C ' 
and d J^±M _ Kav _ pw) = L , &c. ( 12 > 
[Memorandum: — Lines of reference fixed in space], / 
which are more convenient than the Eulerian equations. 
The integration of these equations, when neither force nor 
couple acts on the body (X = 0, &c. ; L = 0, &c.), presents no 
difficulty, but its result is readily seen from § 21 ( u Vortex 
Motion”) to be that, when the impulse is both translatory and 
rotational, the point P, round which the body is isotropic, moves 
