470 
ME. A. COHEN ON THE DIEEEEENTIAL COEFEICEENTS 
numerical quantity, then wP may be used to denote a line which is in the dh’ection of 
P, and whose length bears to that of P the ratio of w to 1. 
6. The following Lemmas, which will be of constant use, are all but self-ewdent: — 
I. If K=(P)— (Q), then (R)— (P)4-(Q)=0. In short, the ordinary rule of signs holds 
good. 
II. (wP)+(^Q)=w|(P)+(Q)}. 
III. The projection on any line or plane of the complete sum or difference of two 
lines is equal to the sum or difference of their respective projections on the line or plane. 
7. Whenever there is no risk of any mistake, the brackets may be omitted in the above 
and similar formulae, and the complete sum or difference of hnes may be spoken of s im ply 
as their sum or difference. 
What has been hitherto said may be of course extended to all magnitudes whatsoever 
which can be adequately represented by straight lines, such as forces, velocities, axes of 
couples, axes of rotation, and accelerations, &c. 
8. The application which can be made in dynamics of this conception of the 
complete differential coefficient of a line, will become at once apparent from the follow- 
ing considerations. 
Suppose a particle to be moving from A to B. Let O be any fixed point. Then the 
particle’s velocity is, according to its very definition, represented in 
AB 
magnitude and direction by the limit of But AB is the com- 
plete difference of O B and O A, or the complete increment of the 
AB 
radius vector O A, and therefore the velocity, being the limit of is the complete 
differential coefficient of the radius vector. 
In the next place let O A and O B in the last figure represent in magnitude and direc- 
tion the successive velocities of a particle at times t and respectively. Then, since 
OB— (OA)-|-(AB), it follows that, if with the velocity OA we compound the velocity 
AB, we shall obtain the velocity OB, and therefore the particle’s acceleration is repre- 
AB 
sented by the limit of or by the complete differential coefficient of the velocity OA. 
Hence we have the following proposition : — 
If a particle’s velocity and acceleration be represented by straight hnes, the velocity 
will be represented by the complete differential coefficient of the radius vector drawn 
from a fixed point to the particle, and the acceleration will be represented by the complete 
differential coefficient of the velocity. Or more briefiy, the velocity is the complete dif- 
ferential coefficient of the radius vector, and the acceleration is the complete differential 
coefficient of the velocity, and is therefore the second complete differential coefficient of the 
radius vector. So that if R denote the radius vector, and V and F denote respectively 
the velocity and acceleration, we have 
