AND DETERMINANTS OE LINES. 
471 
9. Such then being the connexion which exists between the diiFerential coefficient ot 
the radius vector and the velocity and acceleration of a particle, I will proceed to prove 
some of the principal propositions concerning the differential coefficients of lines, and to 
apply them to the dynamics, or rather the kinematics of a particle. 
The first proposition is the following : — 
If P, Q, R represent straight lines, and if we have 
(P)±(Q)=R, 
then 
D,(P)±D,(Q)=D,(R). 
For let P, Q, R after an interval of time become P', Qf, R' respectively, then we 
have 
(F)+(Q0=R', 
and therefore, by Lemma I. of section 6, it follows that 
KF)-(P)i + {(Q')-(Q)} =(K')-(E), 
or 
A(P) + A(Q)=A(R). 
Therefore by Lemma II. of section 6, we have 
/A(P)\ /A(Q)\ A(R), 
A^ Ai J— Al • 
and taking the limit of both sides of this equation, we obtain 
D,(P)±D,(Q)=D,(R). 
Similarly it may be shown that 
i)a(P)+(Q)±(i^)} = WP)+ WQ)±D.(R). 
Moreover, denoting the second complete differential coefficient by D^, it follows that 
i)n(P)±(Q)±(^)}=i).P.(P)+WQ)+WR)} 
=D^(P)±D?(Q)±D?(R). 
10. Suppose now a line Q to have Q^, Q^, Q^, for its components parallel to the axes 
of coordinates O^v, Oy, Oz ; it is evident from Lemma III. of section 6, that 
Q=(QJ+(QJ+(Q*)- 
It follows, therefore, from the preceding section, that 
D,(Q)=I),(Q.)+D,(QJ+D,(Q,) (1.) 
and 
D?(Q)=Df(Q.)+D^(QJ+D?(Q.). (11.) 
These equations are true whether the axes of coordinates are fixed Cr move. But 
supposing the axes to be j^sed axes, let g'^, be the respective lengths of Q^, Q^, Q^. 
Then it is evident that, as the direction of Q, does not vary, D^(Q^) is a line whose 
direction is that of or Ox, and whose length is ^ ; and similarly, D((QJ is a line whose 
direction is that of Ox, and whose magnitude 
Similar results hold good for 
3s2 
