184 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROxM THE CONCEPTION 
and incorporated by actual multiplication. I now proceed to give, generally, the 
mode of applying the notation to determine the conditions of equilibrium of a rigid 
body, or a system of particles. 
(74.) Remarhable symbolizatioji of a Force acting at a Specified Point of a Rigid 
Body. One of the most remarkable symbolizations which my proposed method leads 
to appears to me to be the following. 
Let U denote a force (in magnitude and direction) acting at any assumed origin A, 
then II+M.U 
will completely denote the same force supposed to act at the point u, that is, at the 
point (P) whose distance from the origin is, symbolically, u. For if we apply the 
force at the origin A and then translate it to P, it will be the Fig. 32. 
same thing as if we applied the force directly at P. Thus 
effect of force at P=:etFect at A (or U) 
+ effect of translation from A to P. 
But, the lateral effect only of the translation need be considered, 
for the longitudinal effect is zero, inasmuch as we may suppose a 
force to act at any point of its line of direction on a rigid body. 
Hence the effect of the force acting at P is symbolically represented by the force 
U+ lateral effect of translation, or U+^/.U. 
(75.) Symbol of a Couple . — Let the couple consist of two forces, U 
at the point u, and — U at the point u!'. then these two forces are com- 
pletely represented, as regards their effect on the rigid body, by the 
expression 
(U+«.U)+(-U+«'.(-U)) 
or {u — u').\], 
which is the general symbol for a couple ; as indeed is clear beforehand from the 
fact, that the couple is that which translates the force U from the point {u') to the 
point (m), i. e. along the line {u—u'). 
It will be generally simpler to employ the symbol in the form 
M.U, ' 
u here denoting, symbolically, the line drawn from the point of application of the 
force ( — U) to that of the force (U). The directrix of this, i. e. the axis of the couple. 
is D(m.U). 
(76.) To combine a given Set of Couples. — Let us put (see art. 15) ' 
u — xa + ?//3 + zy 
U = X a -)- Y(3 -|- Z y . 
Then we find, by lateral multiplication, 
u.V={xY-yX)a.^-\-{yZ-zY)^.y-\-{zX-xZ)y .a. 
Hence, if we suppose the given couples to be w.U, m'.U', m".U", &c., and if we 
put, for brevity, 
2(j:'Y-^X)=N, 2(3/Z-;sY) = L, 2(.^X-xZ) = M, 
