474 
ME. A. COHEN ON THE DIEEEEENTIAL COEEEICIENTS 
Again j the complete differential coefficient of the line JL^ to Q) is similarly the 
complete sum of the line [qrs) _J_^ to and of a line whose length is and whose 
direction is perpendicular to the line {qTS to Q), and whose direction is therefore 
evidently opposite to that of Q. Hence I)^(Q) is the complete sum of 
II to and _L^ to and ( 2 ^) J_''to and [—q7^ |j to Q). 
Therefore 
II + X'toQj- 
In other words, the components of Df(Q) parallel and perpendicular to Q are respect- 
ively 
and q’^^-J^{q^). 
15. This last result may be easily applied to the dynamics of a particle. For, let Q 
stand for the radius vector of a particle moving in a given plane. Let that radius vector 
have r for its length and for its angular velocity ; then, since the acceleration equals 
the' second differential coefficient of the radius vector, it follows at once from the last 
section, that the components of the acceleration parallel and perpendicular to the radius 
vector are respectively 
g-m' and w+4 («), or \ (r==>). 
16. This last result is^ however, but a particular instance of the connexion which 
exists between the actual motion of a particle, and its motion relatively to axes which 
move in the same plane as the particle moves. It will be found that that connexion 
may be easily deduced from the solution of the following problem : — 
“ Supposing the axes of coordinates Ox and Oy to move about O in the plane of xy 
with an angular velocity vs at time t, it is required to find the complete differential coeffi- 
cient of a line Q which moves in that plane, the lengths of Q’s components along the 
moving axes being given.” 
Let Q have for its components and Q^, and let the respective lengths of these he 
q^ and q^. Then, by Lemma III. of section 6, we have 
Q=(Q^)+(Qj,). 
Whence it follows that 
D,(Q)=D,(Q.)+D,(Q,). 
Now since and Q^, vary in direction as well as magnitude, and since the angular 
velocity of their change of direction is tjs, we have, by the fundamental proposition in 
section 12, 
F^(Q.)= II to to Ox), 
D,(QJ= II to Oy) +(vsq^ J_^ to Oy). 
(I-) 
