484 
ME. A. COHEN ON THE DIEEEEENTIAL COEFFICIENTS 
The two last terms of this expression are evidently what would be Df(Q) if 
^ were zero, that is to say, if Q did not vary in magnitude ; and \\ to is evi- 
dently what Df (Q) would be if Q did not vary in direction ; so that we have the 
following proposition : — 
“ D|(Q) is, the complete sum of the two partial second differential coefficients obtained 
by varying separately the length and the direction of Q, together with 2 det (Q, QJ, 
where Qi is the line H to 
34. Suppose Q to be E the radius vector of a moving particle, the length of which 
radius vector is r, then D|(Q) is the particle’s acceleration; Qj is ^ |1 toE^, and 
is therefore the velocity along the radius vector. If, then, we denote this by Ej, the 
equation arrived at in the last section shows that the acceleration is compounded of 
II toR)+2det(n,E,), 
and of what would be the particle’s acceleration if E did not vary in magnitude, that is 
to say, if the particle simply revolved about the origin. And this latter acceleration is 
again compounded of det (0^(0), E) and det |0, det (O, E)[. The last line is, by section 
30, a line drawn from the extremity of E, or from the particle, perpendicular to and 
towards O, and whose magnitude is being the length of that perpendicular. 
35. The above result is, however, but a particular instance of the theory of the motion 
of a particle relatively to axes which revolve about the origin, a subject which we are now 
in a condition to treat of very simply in its utmost generality. That theory will be found 
to depend upon the solution of the following problem : — 
“ Supposing the axes of coordinates Ox, Oy, to revolve round the origin O about 
an axis O at time t with angular velocity ts (which is the length of O), it is required to 
find the complete differential coefficient of a line Q, the components of Q along the 
coordinate axes being given.” 
Let Q have for its components Q^, Q^, Q^, and let the respective lengths of these be 
a., q., q,. Then evidently 
Q=(QJ+(Q.)+(QJ. 
Therefore 
D,(Q)=D,(QJ+D,(QJ-hD,(Q,). 
But, by the fundamental formula of section 32, we have 
D.(Q.)= (% II to +det (Q, QJ, 
D,(Q,)= (|t II to Oyj +det (£2, Q,), 
D,(Q,)= (|- II to Oz) +det (£2, Q,). 
