OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
201 
This is the general equation of motion of a rigid body about a fixed point. It 
gives the three well-known equations immediately by equating the coefficients of 
a,(3,y*. But the equation (8.) as it stands is more available for the solution of 
problems, and furnishes results much more simply, than the three equations alluded to. 
Along with (8.) we must employ the equation 
du ^ , 
( 9 -) 
And these two are completely equivalent to the six equations commonly employed. 
(104.) The line represented by the symbol Aa/ia-f-Biy^/S-l-C^yay has a remarkable 
relation to the instantaneous axis which may be thus interpreted. 
Suppose the rigid body to undergo a distortion or unequal expansion of such a nature, 
that all lines in it parallel to a become A times longer than before, all lines parallel to 
/3, B times longer, and all lines parallel to y, C times longer. The effect of this will 
be to convert the unit a into Aa, (3 into B(3, y into Cy ; and thus the line 
will be converted into A&)iC6-{-Bii),(3-\-Ciif3y. This latter line, therefore, I may call the 
Distorted Instantaneous Axis. 
(105.) The distortion here alluded to is one of great importance to be noted, because 
it indicates an operation which has immediate connection with many remarkable 
physical phenomena as well as with various theories in Solid Geometry. As regards 
its geometrical meaning, if we conceive the rigid body to be a solid composed of 
spherical shells having a common centre at the origin, each shell will be converted 
into an ellipsoid by the distortion. The sphere whose radius is unity will be changed 
into an ellipsoid whose axes are Aa, Bj8, Cy ; and the axes of the other ellipsoids will 
be parallel and proportional to these. 
The line represented by the symbol 
Aa-1-B(3 + Cy 
is an important determining element. If we assume cJ to denote what u becomes in 
consequence of the distortion, it may be easily seen that cJ is a distributive function of 
w and Aa-|-Bj3-1-Cy ; and from this fact a number of curious symbolical relations 
may be deduced. But I must noj; dwell upon this subject of distortion now further 
than my immediate purpose requires. 
(106.) Using a’ for brevity to denote the distorted instantaneous axis, 
A^WjCf -j- Biy2/3 “1“ Ciy3y, 
J may observe that the equation (8.), that is, 
^=2mDM.U, (10.) 
* Observing that ^=Da;.a, f^=Da;./3, ^=Da;.v, we find the coefficient of y in the first member to be 
at at dt 
and putting U=Xa + Y/3 + Zy, we find, in the second member, 
'2,m{xY — yX). 
2 D 
MDCCCLII. 
