190 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
Now here, by ( 1 .), M 
is the expression for the momentum of M con- 
centrated into the centre of gravity, and is the momentum of the body as 
regards its motion relatively to the centre of gravity (as is evident from (1.), for, if 
the origin have no momentum, '2m ^ = 0). 
In Section V. I shall show, that 
2m = D“*(A<yi05+ 64/2)3+ CiWay), 
that is, a couple whose axis is As/ja +64/2/3 +04/37. Here A, 6, C are the moments 
of inertia about the three principal axes, of which a, (3, y are supposed to be the 
directions ; and 4/1, 4/2, 4/3 are the well-known component angular velocities. 
(86.) General Expression for the Energy of a Rigid Body moving in any manner . — 
1 venture to suggest the word “ energy' as a proper designation for the total effective 
force by which the motion of a rigid body is produced, inasmuch as evepyeia means 
force actually exerted and effective. The symbol, then, of the energy of a rigid body 
will be 
Now, observing that have 
d'^u ,/ du\ 
Hence the expression for the energy is 
d f..- /du , du 
Comparing this with the expression for the momentum in art. 84, we have 
d {momentum) 
energy = 
( 1 .) 
(870 This, though a very simple result, is really one of importance; thus if the 
centre of gravity be fixed, we find (by art. 85.) 
energy =D“' ^(A4/,a+ 64/2/3+04/37). 
Now it will be shown in Sect. V, that 
7“=D-«, f =D»./3, |=D..r, 
where 4/=4/ia+4/2/3+4/37. 
Hence, performing the differentiation, and observing that 
D‘"'a=/3.7, D"‘)3=7.a, D~'7=a./3, and D"*D=1, 
