OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
189 
du 
Here put for its value, and there results 
^a+/’iyj3=^/3+ constant (c/3' suppose*). 
Multiply this longitudinally by {3, and, since j3xoi=0, j3x(3=l, and |3x(3' = «Xa' 
= cos we find 
a, 
rcu=^-\-c cos 0 ; 
or, since ra=-, this gives 
1 I ^ 
the well-known equation, 0 being the angle Avhich a (the direction of the radius 
vector) makes with a' (an arbitrary constant direction). 
The manner in which the symbolic forms have effected this integration appears to 
me to be worthy of notice. 
(84.) General Expression for the Momentum of a Rigid Body moving in any 
manner . — The Momentum of a particle (m), moving with a velocity is 
du 
And this symbol represents the Momentum in direction as well as magnitude. Now 
momentum is really a force, estimated, however, somewhat differently from ordinary 
pressure, on the principle that the true dynamical effect of a force is proportional to 
absolute intensity and the time of its action conjointly 'f-. Hence, regarding the 
momentum of m as a force, its complete symbol will be (by art. 74), 
Thus the symbol of the total momentum of the rigid body will be 
(>■) 
(85.) If M denote the distance of the centre of gravity from the origin, and the 
distance of the point {u) from the centre of gravity, we have u—u-\-'d, and thus, 
since 2mM'=0, (1.) becomes (putting M for 1m) 
Mf+..(Mf) + 2»(.'.^) 
* Of course the constant is the symbol for some constant line, as regards direction as well as magnitude, 
and therefore I put it in the form cj3', c being a number and (o' a “ direction.” 
t If the pressure P produces a velocity v in the time t, it produces the velocity — per second, and therefore 
t 
V 
P=nij; or 7nv=Vt. Vt then is the momentum, and this is proportional to P and t conjointly. 
