59 
The left member of this inequality consists of / terms, each of 
which relates to an element of volume vj. If we take into conside- 
vj 1 my” 
ration that w=— wy Ef yw, then it is seen that we have 
Vv 
Eet dp 
dn,,? Ll On,2 
and 
SW) lL: udp 
On,,0n,,  ¢ Òn,Òn„ 
The coefficients of all / forms therefore will be the same for all 
corresponding terms. In order to find the condition which is to be 
fulfilled by the coéfficients in (8), we will consider the case 
CLD = QW ov Ox = — On CAT GR 
all other v’s being 0. For this case we have for all possible values 
of the v’s 
only the index 4 occurring. 
The conditions, necessary for this to be true, are that 1. the 
discriminant A 
07 0°y 0° 
On k On,Ong On, dng 
dw dw Op 
ei tao ee ed 
- On, On, dn,? On,0n}. > a 
ow Ow oy 
On, On. Onj-0n}. On} 
whereas the same must be true for the determinants originating 
from the discriminant if we successively omit the right-hand column 
and the last row. The conditions under which the system is really a 
maximum and therefore stable, agree with the well-known thermody- 
namical conditions of stability. 
4. We are now able to determine the mean values of the squares 
of deviations 9?,, and of the products ex» '). 
As is easily seen we have 
ie 07x. e : 7 ° ° ° e e . (11) 
and 
1) Mathematically speaking, our problem is one of correlate probability, my 
formulae agreeing with formulae Prof. J. GC. KAPTEIJN communicated to me after 
1 had solved this problem. 
