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 func- 
tions B,_1(6) that the convergence of their series expansions 
depends essentially on the order of magnitude of T. The small- 
er 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