Atmospheric Currents. • 143 
any fixed line through the centre. If the sphere be rotating 
about this line, its rotation (its surface being supposed fric- 
tionless) will not affect the motion. 
This result is not in contradiction (as Mr. Heath maintains) 
to the fact that the particle moves uniformly in a great circle; 
as the following proof shows. 
Take the particular case discussed by Mr. Heath — that of a 
particle projected from the equator along a meridian towards 
the north. The relative initial velocity here is northward, and 
must be compounded with the westward velocity of the equator 
to get the absolute initial velocity. Let <£ denote the angle 
which this resultant velocity makes with west, and X2 the con- 
stant angular velocity with which the particle moves in its own 
great circle — a circle which cuts the equator at the angle <£. 
Also let X, I denote the absolute latitude and longitude of the 
particle, measured from the starting-point. Then the particle 
in time t describes an arc £lt which is the hypotenuse of a 
spherical right-angled triangle having X and I for sides, and cf> 
is the angle opposite to the side X. 
Hence, by spherical trigonometry, 
cos £lt = cos X cos/, (1) 
tan I = cos <f> tan D,t (2) 
Differentiating (2), we obtain 
dl 
that is, by (1), 
— XI cos ^ sec 2 £lt cos 2 1; 
— - = O cos d> sec 2 X. 
dt T 
This is the angular velocity round the earth's axis. Let R 
denote the earth's radius ; then the distance from the axis is 
R cos X. The linear velocity of rotation is therefore 
Rflcos cf) sec X, 
and the rate of describing area is 
| R 2 f2 cos 4>, 
which is constant. Q. E. D. 
In the meteorological application which is under discussion, 
the moving air receives not merely an initial impulse but a 
steadily applied force directed towards the north, as it travels 
from lower to higher latitudes. Mr. Heath's argument to 
show that it can never reach the polar regions is therefore 
beside the mark. 
In spite of the very bitter tone which Mr. Heath has chosen 
to adopt towards Mr. Ferrel, he is thus clearly wrong in the 
main point of his criticism. 
M2 
