476 
ME. A. COHEN ON THE DIEFEEENTIAL COEFFICIENTS 
If the radius vector be chosen as axis of then x=t^ \ therefore along 
the radius vector, vf=.rw perpendicular to the radius vector, where w is the angular 
velocity of the radius vector. 
20. Let us now apply the same formulae (HI.) of section 18 to the acceleration of a 
particle. Let and as before, denote the components of the velocity, and lety^ 
denote the components of the acceleration of the particle. Then, since the acceleration 
is the complete differential coefficient of the velocity which has Vy for its components, 
it follows at once from the formulae (HI.), that 
dt 
VyX^, 
Suppose now the axis of x to be the radius vector, then we have already shown that 
Vy=r7s. Therefore by substituting these values in the last formulae, we see that 
d^r 
the acceleration along the radius vector, is and fy, the acceleration perpen- 
dicular to the radius vector, is ^(m)-|-CT (wr^), which is the same result as was 
obtained in section 15. 
21. Keturning to the more general case, we have, as before. 
/ » dvj. 
dVy , ^ 
/y— 
dx 
and substituting in these the values already obtained for and Vy, namely, ^ — yar^ 
d^ 
dt 
-{-X'uj, we find 
^ dP-x ^dy „ 
X \ 2 1 ^®' 
(IV.) 
which are the formulae for the components of the acceleration in terms of the coordinates 
of the particle. 
22. The last formulae (IV.) may also be obtained in the following manner. 
Let and represent respectively the components of the radius vector Q along the 
axes of X and y. Then 
Therefore 
Q— (QJ+(QJ- 
D^(Q)=Df(Q.)-fD^(Q,). 
