488 
ME. A. COHEN ON THE DIEEEEENTIAL COEFEICIENTS 
B7,, Tsj-g be the components of vr along fixed axes, and be its components along 
moving axes which coincide with the former at time t, then d^_d^ 
^ ^ dt dt' dt dt' dt dt 
It is evident that this amounts to saying that 0^(0) has ^ for its components ; 
ILl/ xJLL (JLL 
and we have just seen how that proposition follows at once from the fundamental 
theorem in section 35, and from the self-evident fact that D^(0, O)=0. 
41. Recapitulating then the results of the two last sections, we §ee that a particle’s 
system-acceleration is equivalent to det (0^(0), R) minus the centrifugal force, and that 
the absolute acceleration of the particle is compounded of the relative acceleration of 
the particle, its system-acceleration, and 2 det (12, Vi), Vj being the particle’s relative 
velocity. 
If we now look back on the analytical expressions obtained in section 38 for the com- 
ponents of the absolute acceleration, it will be easy to see their full meaning. The 
expression ^ is the component of the relative acceleration. The expression z ^ —y ^ 
is the component of det (0^(0), R), since, as we have seen, D^(Q) has for its components 
The expression 2 is the component of 2 det (O, VJ, since Vj 
has for its components 
d'U7y d'lZz 
dt' dt' dt ' 
Finally, it may be easily shown by analytical geo- 
metry, that the expression 
{y-^z — Z’^y)'^y —{X^z — Z'^x)‘^z 
is the component of the line drawn from the point {x, y, z) on the line whose direc- 
tion-cosines are proportional to p being the length of that perpendicular. 
Hence it is manifest that the analytical formulae in question merely express the proposi- 
tion enunciated at the commencement of this section. 
It may, finally, be observed that the above results might also have been easily deduced 
from the formula in section 34 for the acceleration along the radius vector in exactly the 
same manner as the corresponding analytical formulae for the relative motion of a par- 
ticle in one plane were deduced in section 22 from the formula for the acceleration of 
the particle along the radius vector. 
42. If the origin of coordinates also moves, it is evident that the particle’s actual 
acceleration is the resultant of the acceleration of the origin and the acceleration rela- 
tively to the origin. Hence substituting for the latter acceleration the expression 
already found for it, it is easy to see that the particle’s actual acceleration is, as before, 
the resultant of the relative acceleration, an acceleration represented by 2 det (O, Vj), 
and the particle’s system-acceleration, but that the system-acceleration is now the 
resultant of the acceleration of the origin, and of the system-acceleration relatively to 
the origin, for which latter system-acceleration we have already obtained the expression. 
Now in whatever way a system moves, the motion may be decomposed into a motion of 
translation and a motion of rotation. Hence we see that a particle’s absolute accelera- 
