AND DETEEMINANTS OF LINES. 
485 
Now we know, from section 26 of Chapter II., that 
det (O, QJ+<iet (O, QJ+^et (Q, QJ 
= det{0, 
=det (O, Q). 
Therefore 
®<(Q)= (# II o*) + (|- II Os') + (w li t" O''') Q> 
This formula is true whether the axes be rectangular or oblique, and may be made 
the basis of all the formulae of relative motion. 
It may be observed that, if the coordinate axes did not move, Di(Q) would be equiva- 
lent to 
II toO*) + II to Oi,) + II toO*) , 
So that the line represented by the last expression may be called the differential coeffi- 
cient of Q relatively to the moving axis, or, more briefly, the relative differential coefficient 
of Q. The above formula of the last section therefore shows that the complete differential 
coefficient of Q is the relative differential coefficient of Q together with det (Q, Q). 
This proposition exactly corresponds with the proposition in section 17 of Chapter I. 
36. Assuming now the axes of coordinates to be rectangular, we know, from formulae 
(II.) in section 24 of Chapter II., that det(0, Q) has for its components 
g^THy — gyTss^ parallel to Ox, 
gjss^—g^Tss^ parallel to Oy, 
gy-us^—g^iSy parallel to Oz. 
Therefore it follows from the preceding section, that the components of (Q) are 
dqx 
dqy , 
d>(jz I 
These formulae are in fact simply the analytical expression of the fundamental propo- 
sition in the preceding section, and correspond exactly to the formulae in section 18 of 
Chapter I. 
37. Let us now apply the above formulae to dynamics, and first to the velocity of a 
particle. 
Suppose, then, a particle to move in space in any manner whatsoever, and suppose that 
the rectangular axes of coordinates revolve about a line Q at time t with angular velocity 
t!7, m being the length of O. Let v^, Vy, v^ be the components of the particle’s velocity, 
and Tffy, the components of O. Then, since the velocity is the complete differential 
coefficient of the radius vector E of the particle, and since x, y, z are the components of 
MDCCCLXII. 3 xj 
