110 
[577 
577] 
the pix 
but it 
values 
T1 
conside 
577 
(in fae 
of z, z 
NOTE IN ILLUSTRATION OF CERTAIN GENERAL THEOREMS 
OBTAINED BY DR LIPSCHITZ. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xn. (1873), 
pp. 346—349.] 
The paper by Dr Lipschitz, which follows the present Note [in the Quarterly 
Journal, l.c.], is supplemental to Memoirs by him in Grelle, vols. lxx., lxxii., and lxxiv. ; 
and he makes use of certain theorems obtained by him in these memoirs; these theorems 
may be illustrated by the consideration of a particular example. 
Imagine a particle not acted on by any forces, moving in a given surface; and 
let its position on the surface at the time t be determined by means of the general 
coordinates x, y. We have then the vis-viva function T, a given function of x, y, x', y'; 
and the equations of motion are 
ddT_d r T =0 ddT_dT 
dt dx' dx ~ ’ dt dy dy 
which equations serve to determine x, y in terms of t, and of four arbitrary constants; 
these are taken to be the initial values (or values corresponding to the time t = t 0 ) 
of x, y, x\ y'; say the values are a, ¡3, a, /3'. 
We have the theorem that x, y are functions of a, ¡3, a'(t — t 0 ), (3'(t — t 0 ). 
Suppose for example that x, y, z denote ordinary rectangular coordinates, and that 
the particle moves on the sphere x 2 + y- + z- = c 2 ; to fix the ideas, suppose that the 
coordinates 2 are measured vertically upwards, and that the particle is on the upper 
hemisphere; that is, take z = + f(c 2 — x 2 — y 2 ), we have 
ß '(t- 
T=$(x' 2 +y' 2 + z' 2 ), 
where z' denotes its value in terms of x, y, x, y ; viz. we have xx' + yy' + zz' = 0, or 
V(c 2 — x 2 — y 2 ) ’ 
xx' -f yy' 
Dr I 
functi