AND DETEEMINANTS OE LINES. 
479 
an eye looking along it towards the origin, the revolution of Q towards P appears to be 
a revolution in the positive direction.” 
The determinant of Q to P may be briefly denoted by 
det (Q, P), 
It is evident from the above deflnition, that the determinant of Q to P is a line equal 
and opposite to the determinant of P to Q. 
Moreover, if the projections of P on the axes of coordinates be^^, and those of 
Q be qy, q^,, then it follows from the formulae (I.), that the determinant of Q to P or 
det (Q, P) has for its projections 
SaPz 9.zPy'> 9.xPy 
25. The connexion which exists between the notion of a determinant of lines and the 
elementary conceptions of dynamics may be easily made apparent. For suppose a 
particle at the extremity B of O B to be revolving about the line O A with an angular 
velocity represented in magnitude by O A, then if O C be drawn perpendicular to the 
plane A O B, and equal to twice the area A O B, it is evident that O C will represent 
the linear velocity of the particle. But O C is then by deflnition the same thing as the 
determinant of OA OB. Whence it follows that the determinant of O A. to 
re'presents the velocity of the point B, due to a rotation whose axis and angular velocity 
are represented hy O A. 
This result, together with the result of the preceding section, may then be recapitulated 
in the following manner. If V represent the velocity of a particle at the extremity of 
the radius vector B, and the particle rotates about an axis which is represented by the 
line n, then, if the angular velocity is represented by the length of O, we have 
V=det (n, B). 
Secondly, if P represent a force at the origin and B represent the radius vector at the 
extremity of which a force — P acts, then the axis of the couple (P, — P) is det (P, B) 
or det (B, P) ; so that det (B, P) is what French writers call “ the moment-axis of a 
force P with respect to the origin.” 
26. Such, then, being the connexion between determinants and statical and dynamical 
conceptions, I will proceed to prove some of the more important propositions concerning 
the determinants of lines. 
The most important theorem concerning the determinants of lines is the following : — 
“ If P, P', and Q be three straight lines drawn from the origin, then 
det (P, Q)+det (F, Q)=det {(P)+(F), Q}.” 
This proposition might be easily proved by geometry, but it is at once deducible Aom 
statics. For, consider two couples having a common arm Q, and having forces P and P' 
respectively acting at the extremity of Q at the origin O. The resultant of those two 
couples will be a couple having the same arm Q, and having for its force acting at O the 
resultant of P and P', or (P)+(P'), Now it is proved in statics that the axis of this 
3 T 2 
