AND DETEEMINANTS OF LINES. 
509 
Now the above analytical proof merely shows that the sum of the moments about 
the coordinate axes, of the moving forces of which m det (D^ Q, R) is the type, are 
respectively A B C and that the sum of the moments about the coordinate 
axes of the moving forces of which m det (O, V) is the type are respectively (C— 
(A— (B— 
73. I will next proceed to show how' these results can be obtained far more briefly by 
applying the propositions concerning the determinants of lines. In the first place, if we 
put, for Di(Q), P, and suppose P to have for its components'^, parallel to the axes, 
then m det (Df(0), R) has for its components parallel to z and to 
m{yp^—xpy) and m{xp^—zp^), 
and therefore the moment of m det (0^(0), R) about the axis of x equals 
—{^Pz— ^Px)z \=m{z^ -\-y‘')Px — myxpy — mxzp^. 
Therefore, if we take the sum of these for all the particles and remember that the 
axes are principal axes, that sum wiU equal 'p^1m{z^-\-y^). Now we have already 
proved that the component parallel to the axis of x of 0^(0),=^''. Hence we see 
that the sum of the moments about Ox of the moving forces of which m det (D((0), R) 
is the type is A and that this follows from the properties of principal axes, and from 
the fact that the component of 0^(0) parallel to the axis of x is 
In the second place, we have already seen in section 30 of Chapter II. that the 
moment about the axis of x of det(Q, V) equals — cos <p, being the component of 
\ , and <p the angle between the radius vector and V. 
But —rm^cos<p equals evidently —v^XTS^+yzs^+ZT^,), and equals Z 7 ff,j—yz!!,. If 
then we observe that 2(mxv,)=0, ^myv,)= — ^m 2 f)zs,, ^{mzv^)=^{mz‘^)zSy, it follows 
easily from the above that the sum of the moments about of the moving forces, of 
which m det (O, V) is the type, equals 2(m0^) This 
last proposition may be also proved in a difierent manner, which will show its connexion 
with the proof first given of Eulee’s equations. 
The moment-axis, with respect to the origin, of the acceleration det (O, V) is 
det -jR, det(0, V)[, which, as we see from section 30 of Chapter II., equals a line oppo- 
site to V and of length zsvr cos p ; but it follows from the same section that, since O is 
perpendicular on V, det (O, det (R, V)} equals a line opposite to V and of length 
■asvr cos (p. Hence we have 
det {R, det (O, V)}=det {12, det (R, V)}. 
Let then X denote the operation of taking the complete sum of lines. Then it follows 
from the last equation that 
2m det ]R, det (12, V)} =tm det ^ 12, det (R, V)^ 
= det(12,2mdet (R, V)). 
3 z 
MDCCCLXII. 
