330 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that 
..dx d ( ,dx\ . doc 
dq dt\ dq) dq 
Hence we obtain at once, putting 
T = + & + &)}, 
and supposing x, y, z, replaced by their values from ( 2 ), the typical 
Lagrangian equation of motion 
d 0T _ 0T 
dt dq dq 
That 
d dx dx 
dt dq dq 
follows from the equation 
d dx d dx . d dx d dx . d dx _ 
dt dq dq 1 dq dqr, dq dq k dq k dt dq ’ 
Q (7). 
for here dx/dq is supposed to be the partial derivative with respect to q of 
an explicit function of q v q 2 , . . q k , t, and therefore 
d dx 0 dx 0 dx 0 dx 
dq i dq dq dq l 1 dq 2 dq dq dq 2 ’ 
( 8 ). 
8 . But if the system is non-holonomous this process is no longer ap- 
plicable, and another must be sought. In discussing this question I shall 
no longer refer to the Cartesian co-ordinates of the particles composing 
the system, As a rule, we are given only equations connecting the 
quantities by which the kinetic (and potential) energy is primarily ex- 
pressed with the generalised co-ordinates, and there must be as many such 
equations as are required to express the configuration of the system at any 
instant in terms of these co-ordinates. 
Let the kinetic energy be primarily expressed by the time-rates of 
change u, v, w, . . . , of quantities u, v, w, . . . , which fulfil equations. 
Su = a 1 Sq 1 + a 2 dq 2 + . , 
. . . + ct^qi + + 60^2 ’ • 
• • + e fi s j j 
Sv = 4- b 2 $q 2 -f . . 
• • + d i Sq i +/ 1 0 -S 1 +f 2 ds 2 + . . 
• • +fjh 
For a rigid body u, v, , may be taken as the product of the square 
roots J M, J N, . . . , of inertia coefficients M, N, . . . , into velocity com- 
ponents of the centroid, or the products of angular velocities of the body 
about given axes by the square roots of the proper inertia coefficients for 
this case ; and so for other systems. Thus if T be the kinetic energy we 
shall have 
T = \ (id 1 + v 2 + w 2 + . . . . ) 
( 10 ). 
