AJfD DETEEMINANTS OF LINES. 
489 
tion is in all cases the resultant of the relative acceleration, the system-acceleration, and 
an acceleration equal to 2det(12, V), where O is the axis about which the system is 
turning at the time, and V is the relative velocity of the particle. 
This is the most general form of Coeioli’s theorem. 
43. One of the most important illustrations of the theory of relative motion is the 
motion of a heavy particle relatively to a system which revolves uniformly about a fixed 
axis ; for this includes the case of a falhng body and the pendulum, where the earth’s 
motion is taken into accoimt. 
Suppose then a particle of mass m to have for its actual weight W', and for its appa- 
rent weight W, so that a force — W would keep the particle in relative equilibrium or 
(W\ /W\ 
apparently at rest. Then evidently (“I — (^) equivalent to the particle’s system- 
acceleration. 
Let then the particle be acted on by a force P over and above the weight W', and let 
the particle’s actual acceleration be F, its relative acceleration Fj, its system-acceleration 
Fg. Then clearly 
\m J ' \m J 
But by Coeioli’s theorem 
F=(F0+(F,)+2det(a, V); 
. /W'\ /W\ 
and we have just seen that the system-acceleration F 2 = I — I — ( ir ) ’ follows 
that 
(F,)+2det(Q,V)=(E) + (^): (I.) 
w 
m 
is the apparent acceleration of gravity, and is generally denoted by g. 
44. The above formula is quite general ; but in most cases g may be considered as 
constant both in magnitude and direction, its direction being the vertical direction at the 
point of reference or origin. 
We have then the formula 
^■=(s)+(?)-2de‘{Q.V.) (II.) 
This simple formula enables us to solve easily all problems concerning the motion of 
a heavy particle relatively to a spectator on the earth. The formula shows that the 
relative acceleration is found, just as if the earth did not move, by substituting the 
apparent for the actual force of gravity, and by adding on a force — 2mdet(Q, V,), 
where Vj is the particle’s apparent velocity. 
45. Let us take the vertical downwards as axis of z; let the axis of a? be the horizontal 
line drawn from north to south, and let the axis of g be the horizontal line drawn from 
west to east. Then the equation to the earth’s axis is evidently —^=-4— if x denote 
the latitude of the spectator’s position. 
Therefore cos X, sin X. 
