[T4

PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

B.

— CONSEQUENCES OF THE ASSUMPTIONS

Notation

220. It will be useful for what follows to consider the Laplace
 transform ¢(u) of the function

(220-1)

b(u)= = / Blo) as

vith

— A
dR _, dR,
—— = 09 —
A
R R,
in continuous notation, relation (2) becomes relation (211-1) of the text.
If we put

b, = T (0)
BR -

we have
LR=La+T B(o) L [T7,] +T B(1) L [T7] +. +T B(6) L [TR] +.
=[La+} T 8(6) LT] + X T B(6) L7e
If we suppose that T tends towards zero and that we have
limit [La+E T B(6) LT]=La(t)

and if we put
we finally derive

T

ade

LR = af?) + | 7 8(0) LA, db
0

which is the relation (211-1) of the text.
(*} The meaning of hypothesis (b) is in particular that, at least over
a large range, there exists a valid index of real national consumed income
 R,. This last property corresponds to the general hypothesis I have
made in the first Part (§ 117 and 119) [in particular relations (117-1),
211-1) and 211-2)]

11] Allais - pag. 78