1862.] 177 



parts of that expression the same theorem again, we arrive at the 

 elements of which a line's second differential coefficient is composed. 

 Moreover, since those first and second differential coefficients are the 

 representatives of a particle's velocity and acceleration, we are led by 

 these investigations at once to all the analytical formulae for motion 

 in one plane relatively to moving axes, and are at the same time 

 enabled to give to those formulae a very simple interpretation. 



5. I conclude the first chapter by indicating how all the results 

 thus obtained may be made to flow from the ordinary mode of repre- 

 senting lines by means of imaginary quantities. For instance, by 

 differentiating the expression re & ^^T twice successively, we obtain 

 \* , 1 d 



which expression is evidently compounded of the radial and trans- 

 versal accelerations of a particle. 



6. On passing to the general case of a line moving in space, a new 

 conception has to be introduced one, however, which presents itself, 

 more or less disguised, in almost all the ordinary formulae of statics and 

 dynamics. Let OA and OB be two straight lines ; draw OD perpen- 

 dicular to and equal to twice the area of the triangle AOB ; then OD 

 maybe called "the determinant" of OA and OB, inasmuch as its pro- 

 jections on three axes of coordinates are the simplest determinants that 

 canbe found with the coordinate projections of A and OB. Moreover, 

 if OD be drawn in such a direction that to an eye looking along DO 

 the rotation from OA to OB appears a positive rotation, OD will be 

 called the determinant of OA to OB, and may be denoted by 



det (OA, OB). 



7. The connexion of the determinant, as above defined, with the 

 axis of a couple and with statics in general is self-evident. Nor is its 

 connexion with dynamics less intimate ; for if OA represent in mag- 

 nitude and direction the angular velocity with which, and the instan- 

 taneous axis about which, OB is revolving at time t, then the linear 

 velocity of a particle at B, the extremity of OB, is represented in 

 magnitude and direction by det (OA, OB). 



8. Having thus defined the determinants of lines, I proceed to 

 prove a few fundamental propositions concerning them, which will 

 often be found very useful in abbreviating and giving a clear meaning 

 to complicated analytical work. Moreover, those propositions indi- 



