Rotatory Motion of Bodies. 15 
we may obtain the motion of the system, by applying to m, 
m, m’, the formulz for the motion of a point. 
Let the three lines Om, Om’, Om’, be equal; then mm’ will 
make angles of 45° with Om and Om’; and the tension of mm’, 
which acts as an equal moving force on m and m’, being re- 
solved, it will be seen that the foree on m parallel to Om’ is 
equal to the force on m’ parallel to Om. In the same manner, 
we may obtain two other equations, by considering the tension 
of mm" and of m'm’. And these three equations, when put in 
terms of the quantities which are generally used in expressing 
rotatory motion, give us the three equations of motion. 
Let m, m', m", be referred to Ox, Oy, Oz, three fixed co- 
ordinates. 
The co-ordinates of m are xz, y, 2 
2 
U . , , 
™m L,Y, &; 
” 
m DU. Bs 
and it is easily seen, that these quantities are also 
= cos. mOx, cos. mOy, cos. mOz, 
cos. m’Ox, cos, m’Oy, cos. m'Oz, 
cos. m’Ox, cos. m’Oy, cos. m'Oz; 
(because Om = Om! = Om'= 1). 
We shall then have these equations (Poisson, Tr. de. Mee. 
No. 361*.); 
1. O=aa + yy +52 vty +2 =1 
O= an" +y7" +22"? (a). Also, 2? +y° +27 =1} (6). 
O=a'x" + yy" +22" wy? 4 2 =] 
* And in the same manner we might obtain these equations, 
vy + avy + ay! = 0) ate? +2 = 1 
od “bh a? + a of (@’). y +y” +y"= r (0). 
yetyd +y'2"=0 Pat eta 1 
