196 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
where X, Y, Z represent the accelerating forces, whatever they may be, that are in 
action on the point u. 
(94.) The formation of these equations, here given, affords a good example of the 
use of the symbolic form u.v. but, to illustrate the method more clearly, it will be 
worth while to employ, in some particular problem, the general equation. 
T T ^ -T-W f 
3?=U-2»Dy 
du 
'Tv 
( 11 .) 
which includes the three equations (10.), U denoting Xa+Y|34-Zy. The problem I 
shall choose will be that of the Pendulum Experiment. 
Let QP represent the string at any time t, QO its vertical position, c its 
length ; then 
OQ =:C 7 , OP=M, PQ =cy — u, 
also 
direction of PQ = 
cy — u 
Fig. 37. 
/ 
Let T denote (in magnitude) the tension of PQ ; and then, since the direction of 
T is that of PQ, the symbol of the tension is T - ■ ; to which if we add — gy, the 
symbol of the force of gravity, we find 
V = T3^-gy=(T-g)y-'ln. 
Hence (11.) becomes 
d^U T n I du 
i^ = 0-g)y-7«-2«Dy'.^ (12.) 
Now, for greater simplicity, I shall suppose that u represents a small excursion, 
and c a long string. On this supposition we may regard u as always horizontal. 
Also, if we put for y' its value (7.)? the equation (12.) becomes 
{d% , ^ ‘ jT\ du T ^ m du T 
|^+2nsinroy.;^--Mj+|2» cos/Da. ^-( r-g)y|=0. 
Hence, since ^ is horizontal, is vertical, and Dy.^ is horizontal. The equa- 
tion (12.), therefore, is separated into two parts, horizontal and vertical, which. 
being equated to zero, we obtain 
^-l-2wsinZDy.^-^M=0 (13.) 
(T-g)y=2«cos/Da. J. ( 14 .) 
We have to substitute for T in (13.) its value derived from (14.), which on account 
of the smallness of w, and the fact that T is multiplied by ^ in (13.), gives T=g for a 
first approximation. Hence (13.) becomes 
d^u . 71 -. du q 
;^ + 2MSin®r.'^-fM = 0 
( 15 .) 
