dev elopement of exponential junctions, &c.. 
39 
f (£)=(log. t) n , n being a positive integer, we have 
and consequently, by equation (14), 
| log. ( 1 -J-A ) 0* = 0 . (22). 
in every case but where oc=m, when it becomes 1. 2 .... n. If 
n~i, this becomes 
°=~ r + ±— ( 2 3)- 
in every case but where w=i 
If we take f (£) = -*- , or/* ( /)=f— *, we find in the same way 
l=A*o* — A x ~ l o*+ ..... ,+Ao* (24). 
Again, let f{t) = then will/(/) = sec =J, and as 
the coefficient of 9 2x in sec. 9 is (as Euler has shown )* 
v 2x + i ' ^ 0 )> 
that of F* in sec. - ~ t — will be 
V—i 
f_ , 1*,** + * 2X+1 
-^rr+i c (l ). 
/ 7T * 
which, compared with the expression /( * + a) o 2A , gj ves 
I»2* • • • 2r X 
r — 2# “f * I 
o *** 1 ¥ 1 f 
c(i) S =H)-.kL i±L- 0 2x . (Qt) 
which seems the most compendious form in which this com- 
plicated function is capable of being exhibited in finite terms, 
as well as the most easy of computation in any insulated case. 
If /(0 => ~T=--^ZT > we have / ( f ')= tan -L~, and 
a 1 _2 Q 2X—1 
1.2 ..... ( 2* 1 ) * I-f (X+A ) a 
Cale. differentialis. 2X +* C(i) is used to denote the series 
— -d— + — ! — — &c. 
2JT+I 
j 5 
