310 
PROFESSOR DONKIN ON THE 
in the form of a normal solution of any other similar system 
which may be chosen as the pattern. 
In the most usual examples the function to be chosen for Z is naturally sugg-ested 
by the circumstance, that W presents itself under the form of the sum of two func- 
tions Z-l-O, of which the former, taken alone, gives an integrable system. But this 
is not necessarily the case; and it is worth while to observe that the formulae (65.) 
take a simple and remarkable form whatever Z may be, provided that it be a func- 
tion not containing t explicitly. For then, assuming the “integral of vis viva,” X—h, 
as one of the normal integrals of the pattern system*, the element conjugate to h is 
r (the constant added to t ) ; and observing that Z, in (65.), being expressed in teams 
dZ 
of the elements, reduces itself simply to h, we shall have ^=1?- whilst the differential 
coefficients of Z with respect to all the other elements vanish ; so that, if we put 
«i, Z>i, for the remaining elements, the system (65.) takes the following 
form : — 
^ dbi 
r'=-l + 
dh 
dW 
dui 
(72.) 
This, in dynamics, gives the process to be used in the following problem : “ 2b express 
the solution of any dynamical problem in the form of the solution of any other {involving 
the same number of variables) in which the principle of vis viva subsists.” 
59. As an example of the above process we may apply it to determine the motion 
of a simple free pendulum (not taking into account the earth’s rotation). 
Let I be the length Of the pendulum, and let the mass of the material point m 
placed at its extremity be represented by unity. Also let x, y, z be the rectangular 
coordinates of m, the origin being at the position of rest of m, arid the axis of 
directed vertically upwards. The equation to the sphere described by m is 
x^-{-y'^-\-z‘^—2lz-=0, 
and the force-function U is —gz. 
Hence if we take, as the two independent coordinates, the radius vector f of the 
projection of m on the plane of xy, and the angle 6 between and the axis of x, we 
shall have for the differential equations of motion. 
_ d'W 
du ’ 
dv 
u'=—^, 
dg 
, dW 
d^ 
(A.) 
* See art. 19 (where in equation (29.) is a misprint for b,). 
