PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 297 
log (1+ 
Prop. 6. Uy n= (1 + dy) ( Day 
d 
d Mt aes 
nT eal 
For =e @* eine 
Ux+-n=e Tk ae ne 
v 
= (ts = Lye 5 D (D—1) + &e.)u, 
fe ((1+ =) ew ug 
=a) Ct), 
It must be observed that x is considered constant with respect to 4) in the 
1\ log 1 +Ay) 
3) : 
formula (1+ Had we supposed it otherwise, we must have taken 
account of the differential coefficients of * itself. This would have given the fol- 
lowing theorem. 
Prop. 7. Au,={(l—e~*)-P+D_1hu, 
= {(1+ Ay)~#8O-¢ 91 e-4)—1_ 11, 
d 
For A u,=(e7*—1)u,= Gar (7) *+6&e.) u, 
i 
— (;D TT 9 Pe 1 D@D-1) +e. )w 
= (D+He"+ +542) (D+l)e-# +&e.) Up (B) 
={(l—e-")-O7D_1tu, 
=f{(l—e~*); be A +4,)-1_ in, 
mh ae etal a a 
d 2 
Prop. 8. “ets ug= (line 4 3 (z) +ée.) Un 
a eke 5 (D+2) (D+1)e-*"+ &e.) ue 

=(1l—ne-4))-@+Dy, 
It is manifest that these formulze do not follow the distributive law. They 
cannot, consequently, be applied with any great advantage to the solution of equa- 
tions of differences. We shall exhibit their application only in one instance. 
VOL. XVI. PART III. 4F 

