37 
developement of exponential functions, &c. 
above explained. Pursuing this idea, let us suppose F(£) to 
be decomposable into any number of factors/ (£), f’(t) } f" (t), 
&c., and by executing the same mechanical process on the 
expression F ( l -f- A )o x , we resolve it into 
/( 1 + a)./'(i -j- A'). &c. |o + o'+o" + &c.p. 
A moment's attention to the method by which (15) was ori- 
ginally derived, will convince us that (attending to the proper 
application of the symbols) we are at liberty to develope the 
expression | 0 -4- o' 0" &c. j*, and thus we have the 
equation 
F(i + A)o*=/( 1 + A)/'(l+A') &c. j o+o'+&c. p (16) 
Should any one of the functions / ( 1 -f- A), &c., be of the 
form ( 1 -j- A any term multiplied by o l in the developement 
of j 0 -j- o' -f- &c. j* will acquire the coefficient ( 1 -f- A ) k o\ 
which, being, by (14), the coefficient of V in the dovelope^ 
ment of (1 + £ — 1 ) k , or multiplied into 1.2.3 is evi- 
dently equal to k. Now it is the same thing whether we 
write k ! for (i-|-A)*o* after the developement, or at once 
strike out (1 + A ) k , and for 0 write k previously to it. Hence 
we conclude that 
( i+A)*.F{i+a )o^==/( 1 a )f\ 1 -J-A'j.&c. | &+o-f.o'-|-&c. | ; 
(17) 
where, as before, F(£.) =/(£) .f (t). &c. 
The expression /( 1 -|-a)o* is susceptible of a somewhat 
varied form, deducible from the identical equation 
1 
/(/)=/{(/)”} 
The coefficient of V in the second member of this is equal to 
that of t* in/ { (£)*} multiplied by that is, by (13), to 
