108 
PROFESSOR DONKIN ON THE 
And, in the figure, OT=ip, and <p is measured from T in the direction indicated by 
the arrow, which is also the direction of the rotation about the polar axis. Moreover, 
if the direction-cosines of the axes of x, y, z referred to the fixed axes, be respectively 
a, b, c; a', b', c ' ; a", b", c", we shall have 
a = cos ip cos <p— sin ip sin <p cos 0 
a! = — cos ip sin <p — sin ip cos <p cos 0 
a"= — sin ip sin 9 
b — sin 4 cos <p+ cos ip sin <p cos 0 
b' = — sin ip sin cos ip cos <p cos 0 
b"= cos ip sin 0 
c = — sin i p sin 0 
c' = — cos <p sin 0 
c" = cos 0 
p——0' cos <p— ip' sin p sin 0 
q—0' sin p—ip' cos p sin 6 
r=<p'-\-ip' cos 0, 
hence we obtain the expressions for u, v, w employed in art. 31, viz. 
dh 
u=—=—Ap cos p-\-Bq sin p 
dh n 
v— — = Lr 
dip' 
Jr 
w= — z=—A p sin p sin 0—Bq cos p sin 0-\-Cr cos 0, 
