482 
ME. A. COHEN ON THE DIEEEEENTIAL COEEEICIENTS 
Let, then, the components of P, Q, Q', R parallel to any three axes of coordinates 
be denoted hj &c., See., r^, See. Then, if we denote the components of 
U=det(R, Ql) by u^, Uy, u^, we have, by section 24, 
'^x=iz^y — <lyVz\ (I-) 
and since Q'=:det (P, Q), we have 
qz 9.yP=^ 9.^Py’ 
Hence, substituting the values of q^ and qy in (I.), we get 
Ux=:Px{(lyry + qzrz) - qJ,Vy'^y-\-Pzrz\ 
Now by hypothesis R is perpendicular on Q ; hence 
g_x'r'x+qy'r'y-\-(Lzrz =^ ; 
therefore 
^y't'y + qzrz-^—qxI'x- 
Therefore 
Ux= — qxiPxTx-^-PyTy+Pzrz)- 
But as q and r denote the magnitudes of P and R, it is evident that 
cos (p, 
where p denotes the angle between P and R. Therefore 
%= cos <p. 
Similarly 
Uy= —prqy cos p, 
11 ^= —prq^ cos p. 
Therefore the line U of which %, Uy, are the components is a line in direction oppo- 
site to Q, and whose length equals prq cos p, q being the length of Q. Hence if R be 
perpendicular on Q, then det (R, det (P, Q)) equals —pqreo^ p in the direction of Q. 
It follows from the above proposition, that, if Q' or det (P, Q) represent a force or 
acceleration which acts at the extremity of the radius vector R, and if Q be perpendicular 
on R, then the moment-axis of that force or acceleration about the origin is —^pqr cos p 
in the du’ection of Q, and the moments of such force or acceleration about the coordi- 
nate axes are respectively — pr cos p q^, —pr cos p qy, —pr cos p q ^ ; and as p is the angle 
between P and the radius vector R, pr cos <p=xp^-\-yjPy-\- zp^, if x,y, z be the coordinates 
of the extremity of the radius vector. 
Chaptee hi. 
31. We are now in a condition to treat fully of the motion of a particle in space of 
three dimensions ; and it will be found that the propositions which have just been 
proved concerning the determinants of lines, will enable us to show how all the results 
