

PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. YAO 
But Ott, ae ae 
ad” is not equal to dx 2?. 
In proof of the sufficiency of the principle here laid down, it may be re- 
marked, that both symbols of operation and symbols of quantity are defined or 
characterized by the above laws. The symbols of combination are indeed origi- 
nally framed from arithmetic, but are subsequently generalized, and the basis of 
generalization is obedience to these laws. Thus the symbols + and — are gene- 
ralized by collective symbols the reverse of each other, expressed by the equation 
+a—a=+0=—0; where +0 is arithmetical, or signifies (as an operation strictly) 
increased by 0: x and ~ are ‘ cumulative symbols the reverse of each other,’ ex- 
pressed by the equation x@—+a=x1=-+1; where x1 signifies strictly multiplied 
by 1. These definitions are in perfect conformity with the above laws. And a 
similar remarks applies to the general definition of an index. 
Now certain symbols of operation, although not, like symbols of quantity, 
framed with direct reference to the above laws, do, notwithstanding, satisfy them. 
Consequently, algebraic formule which are results of these laws and of nothing else, 
must be correct forms also when the algebraic symbols are replaced by such symbols of 
operation. 
Section I. Linear DIFFERENTIAL EQUATIONS. 
Preliminary Theorems. 
CY ele 
d Cx ° . ° . d 
5. Since (=) "= ce”, it is evident that if e (=) be any function 
whatever of = we hab have oF (=) eee aj (re. (AY. 
z 
Let uw be a function of z, and suppose it expanded in the form w=3 a,,e”"” ; then 
e""u=3a,e"+")*. and hence 
(=)° Cee Sa (m + ry“ Pees by (A) 
=e 2a, +r)! ee 
=e" 3a, (a +7)*e"* by (A) 
VOL. XVI. PART III. 3R 
