AND DETEEMINANTS OF LINES. 
487 
Moreover we know from section 28, that 
D, det (O, R)=det (D,(0), E)+det (O, D,(R)}. 
But since Di(E,)=:V=Vi+det (O, E), therefore 
det(0, Di(E)) = det(0, Vi)+det det (O, E)[. 
Hence, collecting the above results, and substituting them in the equation (I.), we 
find 
F=Fj+ 2 det (O, Vi)4-det (0^(0), E)+det]0, det (O, E)} . 
It may be observed as to this formula, that if Vi = 0, that is to say, if the particle had 
no relative motion, and moved as if rigidly connected with the axes of coordinates, then 
the two first terms of the last equation would vanish ; and therefore its other two terms 
are what the acceleration would be if the particle had no relative motion, and they 
represent what may therefore be conveniently termed the particle’s system-acceleration. 
French writers have given to this acceleration the name of “acceleration d’entrainement; ” 
it is the acceleration of a point which is in the position of the moving particle, and 
which is supposed to be rigidly connected with the system of moving axes, and I there- 
fore propose to call it “ system-acceleration.” Using then this expression, we have the 
follo^ving proposition : — “ The acceleration of a particle is equivalent to its acceleration 
relatively to a system of axes revolving about a fixed point, together with the system- 
acceleration corresponding to the particle and together with an acceleration equal to 
2 det (O, V,), Vi being the particle’s relative velocity, and O the axis about which the 
system is revolving at time Or, more briefly, a particle’s absolute acceleration equals 
the complete sum of its relative acceleration and of its system-acceleration together with 
2 det(0, V,). Such is the brief expression of Corioli’s beautiful and very useful pro- 
position concerning relative motion. 
40. We have just seen that the particle’s system-acceleration is compounded of 
det(D^(H), E) and det det (H, E)|-. As regards the latter line, it is clear from 
section 30 that it is in the direction of the line drawn from the particle perpendicular 
to and towards the axis O, and that its magnitude is jy being the length of that per- 
pendicular. It is therefore equal and opposite to what is usually called “ the centrifugal 
force.” 
As regards the other line det (D<(0), E), it is of course at once determined as soon as 
is known. Now if be the components of O, it follows from the funda- 
mental proposition in section 35 that 1)^(0) is equivalent to 
(^' il O*) + (‘I- II %) + (If II ‘o O^) H-det (Q, Q). 
But it is clear, from the very definition of a determinant, that det (O, O) is zero. 
Hence we see that the components of D,(H) are This is an important pro- 
position often used in the dynamics of a rigid body, and generally proved by means of 
a good deal of analytical work. It is usually expressed in the following manner. If 
3 u 2 
