20 
MR. LUBBOCK’S RESEARCHES 
axes O x, O y, O z, fixed in space, x p y p z p the co-ordinates of the same element 
parallel to three other rectangular axes O^, O y p O z p fixed in the mass M' 
and revolving with it. Let the line NON' be the intersection of the plane 
x t y l with the plane xy, 
Let the angle N O x = ip, N O x t = <p, and the inclination of the plane x t y { 
upon xy = 0. 
x, — x (cos 0 sin sin <p -f- cos vf/ cos <p) + y (cos 0 cos vp sin <p — sin cos <p) — z sin 0 sin <p 
y t = x (cos 0 sin t|/ cos cf> — cos vf/ sin ({>) + ?/ (cos 0 cos \p cos <p -f- sin rp sin <p) — z sin 0 cos $ 
z t — x sin 0 sin vp + y sin 0 cos i{/ z cos 0 
Let X p Y p Z t be the accelerating forces which act upon the element dm in 
the direction of the axes O O y p Oz p 
f \vf + V) dm — A, ^/°(*, 2 + z . 2 ) dm — B, [xf + 2/,~) dm — C. 
p d t = sin sin 0 d — cos <p d 0 
q d t = cos <p sin 0 d vp -f sin <p d 0 
r d t = d <p — cos 0 d tp 
Cdr + (B — A)p q d t = d tj* \x, Y— y t X t ) d m 
Bdq+(A — C)rpdi=d tHz, X t — x, Z,) d m 
A dp + (C — B)qr dt=z d Z, - *, F ; ) d m 
If the axis of instantaneous rotation coincides with the line O I at any instant 
cos 10*,= , ^ 
\/ p 1 + + r ~ 
cos 10 y t — 
cos I O z, = 
\/ p 2 + q~ + r 2 
✓ P~ + 9 2 + r 2 
sin 
TO aV +<? 2 
V P l + q 1 + r 2 
If z, I L be a great circle cutting the plane x j y l in L, 
r. r COS I O *, P 
cos X, O L = ^ r 
sin 1 O z, \/p‘i + q* 
If the accelerating forces X, Y, Z — 0 and B = A, the integrals of the pre- 
ceding equations are 
C — A , . . s • C — A , . v 
r — n, p = c cos — (n t + v), q = c sin — - — (n t + y) 
A A 
