OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
191 
we find 
energy 
-j~ -j- “f” . 
Lastly, if we perform the lateral multiplication denoted by u. here, we find 
energy =|a^+(C-B)^,^ 3J/3 .y 
~ 7 • a 
— A)&;,<y2^a,f3. 
I need not point out the meaning of this formula in relation to Euler’s equations for 
a rigid body. 
(88.) General Expression for the Vis-Mortua" or “ Dead Pull” on a Rigid Body. 
— 1 may use the almost obsolete word ‘‘ Vis-3Iortua” (which has been so well trans- 
lated by the familiar expression “ Dead Pull”) in the sense in which it was originally 
employed to denote simple pressure or impressed force. I shall therefore designate 
the total mechanical effect of the impressed forces acting on a i-igid body as the Vis- 
Mortua or Dead Pull on that body. If U denote the force acting at the point (u), we 
have, therefore, by art. 74, 
Vis-Mortua—'t({]-\-u.\]) : 
this of course is the same expression as that in art. 77- 
If we put, as before, u=u-\-u!, this expression becomes 
2U-1-m.2U-1-2m'.U. 
Here 2 U-|-m. 2U is the symbol of the force 2U acting at the point (w), and 2</'.U is 
(art, 75) the symbol of a couple. 
(89.) As an example, the result of which I shall require in Sect. V., I shall calculate 
the Vis-Mortua of a rigid body acted on by the attraction of a distant particle m', 
taking the centre of gravity of the rigid body as origin, and a, j3, <y as the directions 
of the three principal axes ; assuming also u' to denote the distance of m', r' the 
magnitude of m', and r that of u. 
Here U is a force acting along the line joining the two points {u) and (^^'), and 
inversely proportional to the square of the magnitude of that line. The line alluded 
to is m' — w, its magnitude is {u' —u)y.{u' — u), and its direction therefore is 
Hence 
Also 
V [u'—u) X {u'—u) 
(u'—u) 
U = mm'- 
{{v!—u) X (w' — m)}S' 
{(«' — m) X {u! — u))~^—{;u! Xu! —2u' Xu + uXu)'^ 
= {r'^ — 2u^ X 
/ Sw X 1 
nearly. 
