AND DETEEMINANTS OF LINES. 
477 
But since the axes of w and y revolve at time t with an angular velocity and since 
Q. and Qj, have x and y for their respective lengths, it follows from section 14, that 
and that 
II toOx}+{i|(®V)rtoOx}, 
D|(Q,)=| II to X' to Oy|. 
Whence it is easy to see that the components of D?(Q), or of the particle’s acceleration, 
are 
(V.) 
which equations are clearly the same as those obtained in the preceding section. 
We shall soon prove similar formulae for the more general case of a particle and axes 
of coordinates moving in any manner whatsoever in space of three dimensions, and 
therefore, in order to prevent needless repetition, we shall postpone the further discussion 
and complete interpretation of the equations (IV.) or (V.). 
23. It is, however, interesting here to observe that all the results already obtained may 
be readily deduced from the principles of what Professor De Moegan has called “Double 
Algebra.” According to those principles, namely, the radius vector R whose length is 
r and whose inclination to a fixed line is d, is symbolically represented by so that 
we have 
Therefore 
and the last expression represents the complete sum of 
(5 II to r) and {r~±;io^. 
This result is the same as that arrived at in section 12. 
Again, Df(R)=D,(D,(R))= 
and this expression represents the complete sum of a line 
dfl-^t II to E and a Ime * + X' to E. 
This result is the same as that arrived at in section 14. 
MDCCCLXII. 3 T 
