832 
P, =k (M4: Sr ae Fy Pra Py CHE .+¢)) (i= 100.1, Be. 2) 
Introducing the differentials de, ....dc, instead of the differentials 
dp,.... dp, into the phase-integral 
Laffer ef dood dad sel add BQ) 
we find 
I deed d dp d shi ee 
= ffe, … de ffe dp, Ee TEN ) 
r=ff dy... dolen) sn ze HEAO) 
Ò(p,,---p‚) 
hsl fig +0 RE PETER CN mere LIE) 
In (11) the integration has to be carried out, the limits being 
determined by (8). From the (p;,q;)-space or the (c;, f;)-space we 
may pass to the k-dimensional (c;,...cj)-space. A streamline of the 
former space corresponds to a fixed point in the latter. The density 
@ is replaced by ow in the c-space. Its elements therefore have the 
weight w. For the iso-parametrie motion (a = const.) the c-space is 
static i.e. each point is fixed. The integral (11) gives us the numerical 
mean looked for, namely, if f is a ee. we have: 
0(p,.-- 
oe bran Era -dqg/m. + + . (11) 
Returning to equation (7) we now ren 
where 
k do De. 
Be en eat aT ee yl UE 
since 9 is a function of c,,...,cx only. Similarly the quantities 
Deil ,, only depend on c,,..., Cz, as is easily seen from (6), if on 
the right hand side we replace the time-mean by the numerical 
mean (11'). Therefore 9 retains its property @ = @ (¢,-- ., Ck) when 
a changes. Equation (12) expresses, that @ is a function of those & 
integrals of the differential equations (6) which only contain the 
quantities c,,...,cz. These integrals are obtained by integrating the 
set of & differential equations which on the left side contain the 
quantities De;/,, (¢=1,2,...,4). They are the essential adiabatic 
invariants, and we have thus proved that @ is a function of the 
essential adiabatic invariants. 
