AND DETEEMINANTS OF l.INES. 
473 
the latter line {q-m to Q) being in the plane in which Q is moving at time t. This is 
the fundamental proposition concerning the differential coefficient of a line, and may be 
stated in the following form ; — 
The com'plete differential coefficient of a line Q, whose length is Q and whose direction 
is at time t varying with an angular velocity ro-, is the comffiete sum or is compounded of 
two lines, one being ^ in the direction of Q, and the other being q^n in a direction perpen- 
dicular to Q and in the plane in which Q is moving at time t. The former of these two 
lines would e\idently be the complete differential coefficient of Q, if the length of Q only 
varied, and the latter would be its complete differential coefficient if the direction of Q 
only varied ; and in this sense it may therefore be said that the complete differential 
coefficient of a line is the complete sum of the two partial differential coefficients obtained 
by varying separately the length and the direction of Q. One of these partial differential 
coefficients may be called the length-differential coefficient, and the other the direction- 
differential coefficient of Q, and the complete sum of these tw'o constitutes the complete 
differential coefficient of Q. 
13. Let Q in the preceding section stand for the velocity of a moving particle. Then 
Di(Q) will be the particle’s acceleration, q will be the velocity v, and the direction of Q 
will be that of the tangent to the particle’s path. Finally, rsdt will be the angle between 
ds 
two consecutive tangents, so that zudt——, ds being an element of the particle’s path, 
and P the absolute radius of curvature. Therefore It follows then at once 
§ at g 
from the last section, that Di(Q), the particle’s acceleration, is compounded of ^ along the 
tangent, and vtst or — perpendicular to the tangent and in the plane in which the radius 
vector is moving at time t. In other words, the resolved part of the acceleration along 
dv d^s 2 
the tangent is and the resolved part along the absolute radius of curvature is y. 
14. I. he same fundamental proposition of section 12 enables us to investigate Df(Q), 
the second complete differential coefficient of a line Q, if we suppose that line to move 
always in the same plane. We have, namely, 
D.(«)=(| II toQ)+(,^jj.'toQ). 
Now, in order to find I)f(Q), we must ascertain the complete differential coefficients of 
II to Q)- The complete differential coefficient of the former line 
{ft II section 12, 
(§l|to«)+^(|rtoQ). 
