198 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
V. Application of the Symbolic Forms to determine the INIotion 
OF A Rigid Body about its Centre of Gravity. 
(96.) The symbolic forms u.v and uxv are singularly useful, as it appears to me, 
in all cases of the Motion of a Rigid Body in space, especially as regards Rotation. 
Considerable simplification is also gained by employing da, d^, dy to denote the an- 
gular motions of the three axes a, (3, y. I shall now proceed to consider this case. 
I shall take a, (3, y to denote three rectangular directions fixed in the Rigid Body, 
and X, y, % the coordinates of any particle (m) of the body. On this supposition x, y, % 
are constants as regards t, while a, j3, y are variables. The origin is the fixed point 
(the centre of gravity, namely,) about which the body moves, ii denotes the distance 
of m from the origin, and therefore 
u—xa-\-y^-\-%y (1.) 
(97.) Now the rigidity of the body requires that the velocity of m shall be at 
right angles to u always ; this may be expressed (see art. 44) by putting 
du 
dt 
=-Y)a.u, 
( 2 -) 
where u denotes some unknown line. It may be shown, as follows, that a is a func- 
tion of t only, or, in other words, that a is the same for all points of the body, i. e. for 
all values of u. 
Let {u') be any point in space, and let ns assume, as we may, that this point moves 
always with a velocity D^y.w'; then 
and hence, by (2), 
du' „ , 
(3.) 
d{u'—u) 
dt 
=Di!y.(M' — u) 
(I-) 
Now m'— M is the line joining the two points (w) and {v!), and the square of its 
length is 
(u'—u) X {uJ-^u), 
and m)x(m' — m)}=2^^^^^^X(m'— m) = 0 ; 
for (4.) shows that — and v!—u are at right angles. Consequently the length of 
the line n! — u is invariable. 
In precisely the same way (3.) shows (what indeed is otherwise obvious) that the 
line is of invariable length. 
Hence the point {u') is rigidly connected with the origin and with the point (u) ; 
and consequently (u') is a point of the rigid body. Therefore, comparing (2.) and (3.), 
it appears that m does not vary when we pass from one point to another of the rigid 
body. 
