478 
ME. A. COHEN ON THE DIEFEEENTIAL COEEEICIENTS 
Finally, in order to obtain the formulaB for relative motion, we have merely to put 
where 6 is the angle made by E with the moving axis of x, and a is the angle made by 
that mowng axis with a fixed line. It follows then that 
Now it is evident that ^ ) represents tTw relative dAfferential coefficient of 
E, andr^-v/ — 1 represents a line _L^ to E. We thus obtain the same 
result as in section 16. 
By differentiating again it would also be easy to deduce the result of section 20, if we 
observe that s* ^ \n^ ) represents the particle’s relative acceleration whose com- 
ponents are and 
Chaptee II. 
24. In order to extend the formulae which we have proved for the motion of a particle 
in one plane to the motion of a particle in space, it will be found very convenient to 
make use of a conception which presents itself in statics, as soon as the equilibrium of 
a solid body is treated of in that science. 
Let O A and O B be any two straight lines drawn from the origin O. 
If then O A represent a force P, and if we apply at B a force — P, we 
shall obtain a coujple. Let O C be the axis of that couple. We know 
then from statics that, if O A and O B have for their projections on the 
axes of coordinates X, Y, Z and x, y, z, then O C has for its projections 
zY-yZ, x7.-zK, ylL-xY (I.) 
Now the relation which the line O C bears to the lines O A and O B is one which not 
only presents itself in statics, but which also plays a very important part in the differ- 
entiation of lines, and in the dynamics both of a particle and of a body. For this reason 
it will be proper to treat of the relation in question quite independently of statical 
considerations ; and since the expressions (I.), which are the projections of O C, are evi- 
dently what are called determinants, I shall call the line O C the determinant of to 
OK. 
Hence we have the following definition : — 
“ The determinant of a line Q to a line P is a line which is equal to twice the area of 
the triangle of which the lines P and Q drawn from the origin are sides, and which is 
perpendicular to that area, and the line is moreover drawn in such a dmection that, to 
