( Til ) 
metical mean, so half the sum of the lowest and the highest tem- 
perature; and the second part, viz. 2 7 (26B— B + X)V-X denotes 
the amount that the really occurring temperatures lie above or below 
this middle value. This second part is imaginary outside the limiting 
values of x. For between these limiting values of 2, X is positive, 
and beyond them negative — but the first part exists over the full 
width. The course of this first may be given in the main points. 
Beginning with 7'=0 and «=O it ends also with this value at 
2—=1. But for very small value of z or 1—~., provided it be 
outside the limiting values of «, this first part is negative. 
For the limiting values of 2, where Y —0, it has the above treated 
a 2 
positive value MRT = OIB 
values of a a value equal to 0 must occur; we conclude to this by 
noticing that if # or 1—, is very small, 5? and XB may be neglected 
by the side of B, while X is negative beyond the limiting value of 
x. The curve which represents the first part begins with an ordinate 
equal to zero, then descends below the axis, but intersects the axis 
again before the smallest value of x is reached for which X is equal 
to zero, then rises to a maximum value, after which it descends 
below the axis, and finally terminates with a value zero. 
So if we draw the curve 7}, as in fig. 39, this curve is of course 
the limit above which 7’ cannot rise for any point of the closed 
curve. The closed curve being the section of two surfaces which 
have each a “contour apparent” on the 7'z-plane, the projection of 
the sections cannot fall outside this outline. So the 7,2-projection can 
have either one or two points in common with the curve 7%, in which 
But just beyond these limiting 
; v 
points it must touch this curve. In these points of contact <= 3. 
os 
If there are two points of contact De > 3 between these points. 
v 
The observation that eS 3 in the points of contact enables us to 
show that this circumstance cannot occur for low value of n. First 
of all not for n <2, because, as we saw before, v must there be 
smaller than 6, = 2,. If we introduce into the equation: 
6 fe 2 de RO 
b b 
the condition In 3, we get: 
| 4=-9A-—B. 
48 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
