AND DETEEMINANTS OF LINES. 
483 
arrived at as to a particle’s motion in one plane may be extended to motion in space 
generally. 
32. Suppose Q to represent a line drawn from the origin, varying both in direction 
and magnitude in any manner whatsoever, and let it be required to investigate D^(Q) the 
complete differential coefficient of Q. 
Let the length of Q be at time and let the direction of Q be revolving at time t 
about a line whose direction is that of the hne represented by 12, and let the length of 
Q be the angular velocity zs-, vrith which Q’s direction is revolving at time t. 
It has been akeady shown in section 12 that D^(Q) is in all cases the complete sum 
of the two partial differential coefficients which are obtained by varying separately the 
length and direction of Q. Now the former partial differential coefficient is evidently 
II to Q^, and the other partial differential coefficient is, by section 25, equal to 
det (Q, Q). Hence we have the following fundamental equation, 
D,(Q)=(| II toQ)+aet(Q,Q). . . (I.) 
33. It is not difficult to deduce from the last equation the expression for I)|(Q), the 
second complete differential coefficient of Q. In order to find that expression we must 
take the complete differential coefficient of each of the expressions of which the right- 
hand member of equation (I.) is composed. For this purpose represent for a moment 
the line || to by Qj. Then it follows from the fundamental formula of the pre- 
ceding section, that 
DKQ)=D,(Q0-f D, (det (O, Q)). 
Now the formula (I.) of the last section gives evidently 
D,(Q.)=(J« II to Q,) +det (O, Q.), 
or 
^^2 II to -j-det (O, Qi). 
Moreover we have, according to section 28 of the preceding Chapter, 
D,|det(Q,Q)}=det p,(0), Q[ -f-det |0,H,(Q)}. 
But since 
D,(Q)=:(QJ+det(0,Q), 
it follows from section 26 of the preceding Chapter, that 
det (Q, D,(Q))=det (O, Q,)+det |0, det (Q, Q)[ . 
Therefore, collecting the above results, we obtain 
DKQ)==H,(Q,)+D,|det (O, Q)[ = 
II Q) +2 <let (H, QJ+det (D,(Q), Q)+det{Q, det (O, Q)} . 
