186 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
the former acting at the origin, and the latter at a point whose distance from the 
origin is 
G 
sin $ 
drawn in the direction a'. 
Rp G, y' and y^ are given by the equations (1), (2), (3), (4) above ; a' is determined 
because it is at right angles to the plane and |3', because it is in the same 
plane, and at right angles to y'. As regards we have, taking the longitudinal pro- 
duct of (2.) and (4.), 
LX,+ MY, + NZ, 
LR, 
= y'xy,= cos d. 
(79.) Centre of Parallel Forces . — Let the magnitudes of the parallel forces be 
R, R', R", &c., y^ their common direction, and u, w', m", &c. their points of application. 
Then their combined effect is 
2(RyJ + 2(M.Ry,) = (2R)y,+-^-(2R)y,. 
Now, by art. 74, this is the symbol for a force 2R, acting in the direction y,, and 
at a point whose distance from the origin is 
SRm 
2R’ 
SRa; SRy„ , 2R^ 
SR °‘+^Rr^+lRy’ 
which expresses the common formulae for the centre of parallel forces. 
III. Application of thk Symbolic Forms to Dynamics. 
(80.) Effective Force, Vis-Viva, IVork. — If u be the distance of a moving particle 
m from the origin at any time t, it is clear that du represents, in magnitude and 
direction, the space described in the time dt-, and thus the complete symbol of the 
velocity becomes 
du 
dt 
Also dif) 
represents, in magnitude and direction, the alteration of velocity in the 
time dt ; and thus the complete symbol of the effective force is 
d^u 
m-jTo' 
Again, if U denote any force acting on m, it is easy to see that the symbol of the 
work accumulated, while m is moving from the point u to the point w', is 
U X du. 
Lastly, if for U here we put the effective force, the effective work will be 
dhi 
