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ME. A. COHEN ON THE DIFEEEENTIAE COEEEICIENTS 
Therefore the two equations (I.) and (II.) evidently show that the 
components of D^(Q) andDi(Q) parallel to Oa7are respectively equal to ~ and^^- 
11. It is easy to apply the above results to the velocity and acceleration of a particle. 
For let Q in the last section stand for the radius vector of a moving particle, then the 
components of the radius vector are respectively equal to x, y, and z ; and since the 
velocity is the complete differential coefficient, and the acceleration is the second complete 
differential coefficient of the radius vector, it follows from the last section that the com- 
ponents parallel to Ox of the velocity and of the acceleration are respectively equal to 
doc d^oc 
-yy and So that if be the components of the velocity, and/^,/^,,/^ be the 
components of the acceleration, we have the elementary formulae 
dx dy dz 
p d'^x p d'^y p d’^z 
and similarly 
12. Our next proposition will arise from investigating the complete differential coeffi- 
cient of a line Q, which varies both in magnitude and direction with the time t. 
Let Q at time t be the line O A, and let it become O B at time 
so that we have 
A(Q)=(0B)-(0A)=AB. 
Produce OA to C, making OC=OB, and draw BC. Let OA 
and OB have for their respective lengths q and Ag*, and let angle B O A=05. 
Then 
A(Q)=AB=(AC)+(CB). 
Therefore, by Lemma II. of section 6, 
W 
Now dimmish Lt indefinitely and take the limit of the last equation. The limit of 
AfQl AC 
is Di(Q), the complete differential coefficient of Q. The limit of is evidently 
AC da 
a line whose length is the limit of or and whose direction is that of OA or of Q. 
CB 
Finally, since COB is an isosceles triangle, has for its limit a line whose length is 
OA limit of ■^, and whose direction is perpendicular to OA or Q, and in the plane in 
which Q is moving at time t ; and if w be the rate at which the direction of Q is varying 
at time t, w=limit of Therefore, taking the limit of equation (I.), we obtain 
D,(Q)= (I II to q) +(j»r to Q), 
