PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 249 



But xdx-=1x'^, dxz^ = 3z^ 



xd3^ is not equal to dz x^. 



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 : x and h- are ' cumulative symbols the reverse of each other,' ex- 

 pressed by the equation y.a-=ra= xl =-^1; where x ] 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 s3Tnbols of quantity, 

 framed with direct reference to the above laws, do, notwithstanding, satisfy them. 

 Consequently, algebraic formulae 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. 

 Preliminai^y Theorems. 



5. Since (^)'' *'^= '^ *"> i* i^ evident that ^^ f \j-^ be any function 

 whatever of ^, we shall have J^ (j^j e"" =y (c) /^ . (A). 



Let M be a function of a-, and suppose it expanded in the form « = 2 «,„ <■'" ' ; then 

 e"M=2a^e'"'+')^; and hence 



— \ . e"" u^l. a^^ (m + ry i- , by (A) 



rx ( d \ K. _ mx 



\dx ) •" 



VOL. XVI. PART III. 3 R 



