388] 
NOTE ON THE COMPOSITION OF INFINITESIMAL ROTATIONS. 
25 
where l, m, n, p, q, r are constants depending on the infinitesimal motion of the 
body. 
Hence, first, for a system of rotations 
®i about the line (a 1} b lt c 1} f u g lt hfi, 
>> » (^2? b 2 , C-2) f%> p2> h 2 ), 
&C. 
solid 
the displacements of the point (x, y, z), are 
8x = . yXeco — zXbco + S/ft), 
8y = — x'Zcw . + z%a 0) + 2#ft), 
8z = xXbw + ylZaco . + 2/ift); 
and when the rotations are in equilibrium, the displacements (fix, 8y, 8z) of any point 
(x, y, z) whatever must each of them vanish; that is, we must have 
2ft)tt = 0, Sft)6= 0, S«c = 0, Sft)/ = 0, ’Swg = 0, Hwh — 0, 
which are therefore the conditions for the equilibrium of the rotations co 1 , <w 2 , &c. 
Secondly, for a system of forces 
Pi along the line (oq, b lt c ly /, g x , hfi 
P 2 „ „ (a 2 , b 2 , c 2 , /0, g 2 , h 2 ), 
&c. 
the condition of equilibrium as given by the principle of virtual velocities is 
2P (al + bin + cn+fip + gq + hr) = 0 ; 
or, what is the same thing, we must have 
2Pa = 0, tPb = 0, tPc = 0, SP/= 0, XPg = 0, SPh = 0, 
which are therefore the conditions for the equilibrium of the forces Pi, P 2 , &c. 
Comparing the two results we see that the conditions for the equilibrium of the 
rotations «!, tw 2 , &c. are the same as those for the equilibrium of the forces P 1} P 2 , &c.; 
and since, for rotations and forces respectively, we pass at once from the theory of 
equilibrium to that of composition; the rules of composition are the same in each case. 
Demonstration of Lemma 1. 
Assuming for a moment that the axis of rotation passes through the origin, then 
for the point P, coordinates (x, y, z), the square of the perpendicular distance from 
the axis is 
= ( . — y cos 7 + £ cos /3) 2 
+ ( x cos 7 . — z cos a ) 2 
+ (- x cos /3 + y cos a . ) 2 , 
C. VI. 
4