322 Mr. Landen’s Invejtlgation of 
ab (or a circle) whofe center is A ; fe mi- axis A a -=zn ; and the 
x n ; except c b e~d'; in which cafe the 
projected track will be a right line a h parallel to AC. 
With regard to the permanent axes of rotation of our paral- 
lel opipedon, it appears, by my Mathematical Memoirs , that if 
two of its dimenfions be. equal (that is, when the body is a 
fquare prifm ), any line palling through the center of gravity of 
the body, in a plane to which the other dimenfion is perpendi- 
cular, will be a permanent axis of rotation ; as will the line 
palling through that center, at right angles to that plane. If 
all the three dimenfions be equal (that is, when the body is a 
cube '), any line whatever palling through the center of gravity 
of the body will be a permanent axis of rotation. 
It is obfervable, that the momentum of rotation of the 
body, about the momentary axis, is found by computation 
always = 4- X b~m L + c L dr + d 2 n 2 r e denoting the angular velocity. 
But 4 x frm 2 -f c z d L -f d 2 n 2 is the initial momentum of rotation. 
a x 
Therefore, conlidering the momentum of rotation as invariable, 
the angular velocity will be invariable, e being always =f 
which here denotes the initial angular velocity. 
Our next bufmefs is to find the length of the track defcribed 
by the momentary pole (P), upon the fpherical furfacej and 
the velocity of the pole in that track. 
Fig. 2, 3. It appearing, that the motive force E is = 
Mi? 
- x D in - Ca : x - , and the motive force E — ^4“ x 
cy 
X 
%/ On 2 ' — By ; we find F = \/e 2 + E 2 (the force compounded 
of thofe two forces) = x s/Td 2 mnr — BQa z y z ; and, F being 
to 
3 «‘ 
