2304

PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

(510-10)

wW + WU 40000 + ww _,+...

(510-11)

J

"7p, (0)d0=k

The function 3(0) then appears as the weighted average of
the functions B,_, (0), each satisfying the condition that its
integral over the range zero to infinity is equal to k, and that
the sum of the weights is unity.
Naturally we have

(510-12)

B =k

Thus, the larger w,, the more closely B(8) approximates to
an exponential form.

Convergence of the Series Exbansions

It can be seen directly from the expression for the functions
 B,_1(6) that the convergence of their series expansions
depends essentially on the order of magnitude of T. The smaller
 T, the more rapidly the exponentials converge.
From this point of view, the order of magnitude of the
w, , is of secondarv importance.

11] Allais - pag. 208