( 855 ) 
t 
. | u,(a)dr 
oe 
| ld 
a 
then this number, because w,(7) has a definite sign in the range 
of integration 7, (v), is positive and smaller than unity. 
The numbers 
Bo: Bptis Dye: 
are not ascending, for we have 
t t 
fico dx { Unti(a) der 
ps a a 
ae Bn = 5 b ae a =a 
fw (4) dy | Unti(y) dy 
a a 
or 
t b 
à Uy Di } x 
di dy ul) ul) Ik uy) int i(@) 
, uw) Un{ x) 
t 
Bn | Pn+1 os = b a PEHA Fr ’ 
fan dy . fem dy 
a a 
and as in the double integral y always remains larger than 2, the 
double integral can in connection with the supposition just made 
concerning the fourth datum never be negative. 
Again we can apply the lemma of Ager, from which ensues that 
b t 
oo oo 
= Bs fralydey ==> bald 
p Pp 
a a 
is lying between 3,G’ and 3,4’, where G’ and A’ denote succes- 
sively the upper and lower limit of the sums 
b b b 
| uply)dy, frans + st dekujdijnnn x 5 
a a a 
b b b 
fans +f uptily)dy + J uppe(y)dy de 
u a a 
56 
Proceedings Royal Acad. Amsterdam. Vol. XI(L. 
