347 
1908-9.] On Lagrange’s Equations of Motion. 
of the system with angular velocity 0 sin d about OE, turning towards 
the instantaneous position of OD. But associated with OC is angular 
momentum — C(1 — cos d)0 + C '{\js — 0(1 — cos d)}, where (as explained above, 
§ 20) C refers to the pendulum apart from the fly-wheel, and C r to the 
fly-wheel. Hence this motion gives a growth of angular momentum about 
that instantaneous position at rate 
[ - C(1 - cos 0)4> + C '{0 - 0(1 - cos $)}]</> sin 0 . 
Again, the axis OE is moving away from the instantaneous position of 
OD with angular velocity 0 cos d, and there is associated with OE, in this 
position, angular momentum A 0 sin d. Hence angular momentum about 
the instantaneous direction of OD is growing from this cause at rate 
— A 0 2 sin d cos d. 
From all these causes combined, the rate of growth of angular 
momentum about the fixed direction with which OD coincides at the instant is 
AO - { A cos 0 + (C + C')( 1 - cos 0 ) } sin 0 . </> 2 + C '00 sin 0 ; 
and this equated to L, the moment of the applied forces about OD, gives 
the d-equation of motion. It will be seen that this result agrees with 
(50) above. 
In precisely the same manner the other equations of motion of the 
pendulum could be obtained. 
It will be noticed that the terms which (apart from Ad) are obtained 
in these two examples from the motion of the axes are those contributed in 
the Lagrangian method by the term — dT/dO. This affords an interpretation 
of the term in question, and of corresponding terms in the equations of 
motion in other cases. [See also § 19.] 
31. I may notice here the fact, which I have pointed out elsewhere, 
that this simple principle enables Euler’s equations of motion for a rigid 
body turning about a fixed point to be established intuitively and without 
analysis. Refer to principal axes passing through the fixed point and 
turning with the body. Let A, B, C, be the moments of inertia, and 
<*)]_, ft>2, O3, the angular velocities about these axes. Angular momentum is 
growing about the first axis at rate A d> 1 , in consequence of the rate of 
change of w r But as the body moves the third axis, about which the 
angular momentum is C co 3 , is turning towards the instantaneous position 
of the first with angular velocity w 2 about the second, and hence angular 
momentum is growing about the first direction at rate Co ) 2 a> 3 . Also the 
second axis, about which the angular momentum is B a> 2 , is turning away 
from the instantaneous position of the first owing to the turning with 
angular velocity o) 3 about the third, and hence angular momentum is 
