developement of exponential functions , & c. 43 
F(MO =F(MT*/,(M') 
we shall have 
F| [ X + A ), ( 1 + A') 1 0 *. of) — 
=/{(l+A),(l + A')}x/''{(l+A,),(l + A/)}(o+0,)“ r -( 0 '+ 0 /y> 
• • •• • • • ( 3 l). 
This, as well as other analogous theorems, flows with such 
facility from the principles above laid down, that it is unne- 
cessary, as it would lead me beyond the limits I proposed, to 
enter into any detail respecting them. 
Let us now consider the developement of a function of the 
form f^ n (t), f, and ^ being functional characteristics of a 
given form, and denoting the result of n — 1 successive 
substitutions of if/(£) for t in the expression of vl/ ( t ). Let us 
then suppose, for brevity’s sake, t]/ (log* (l-TO) = ^ (0> an d 
equation (3) will give 
JHt) a )/'* (/) 
and for f writing f\ \ n 1 
again for f writing/^ n ~ 2 and for t, <p (A) we get 
’ /+"-*{?( A)} =/^”- 2 {?(A')}/'« A) 
and so on to 
/+ { * ( } =/* (Af- 0 ) a( " _2) ) 
Collecting, now, the whole result, and, for the sake of con- 
venience, inverting the order of the accents, we obtain, 
f-ty n (0 =/ ( A)£°*? > (A') + o'.?)(A") + — r<>( n l \t • , . . .(32) 
The second number of this equation, actually developed be- 
comes 
/<P(A)o“ {<?> (A) 1*0 13 
-n.n . W ^ «■ - ; X 
I • £»»••&& I.*2 1 • » ^ 
G 2 
