IN THE DIFFERENTIAL CALCULUS. 4] 
=n (d,)u+d, 0 (d,) ut sas, op" (d,)ut be. 
where ¢’ (d, ) is the differential coefficient of ¢ (@, ) with respect to d,. 
5. In Art. 3, let o=e*” , then 
LENIN Se 211 PRTC oe 
+ AUB) (Br de w+ he) 
=e" (4.48 ru; 
and hence, generally, as in Art. 1., 
o(a)ie* ae 
6. Let $(z)=3Be*”, then 
$ (2) ¥(d,)u=3 Bee” y (d,) w 
=3 By (d,—Br) ew by (5) 
« 
a 
=3Bip(d,)—4(a,) Br+¥' (@) 2S — 80.4 Pu 

= (d,) {3B *"u}—¥ (d,) fd,(2Be®™)u} + &e. 
=¥(4,) {f () u}-¥ @,) {4 @®) 
1 
+7 4) td p @) - u}— &e. 
7. It is easily seen that, if m be a whole number, 
ra eam 1/1 1 aL ) 
4," log 2= FF {log 2-5 (Gr5+éer 5) 5 
and that in other cases, 

ar 
df log 2 =(—*)" "faz 
=—m ' r—1 m 
8. The operation denoted by ¢, must not be confounded with (« i d r) ; 
it is, in reality, ( fe ae)". The one expression can be deduced from the 
other, thus :— 
oe aa aap et) 
(Cam dz) "log a liar Mie rc log z= ies 

(=n (E (cre) P- a8) ! bem 
P [=e a 
Te r 
where p, q, and Fs are indefinitely small. 
