YAS PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
tions, it is, perhaps, necessary to say a few words. The principle on which that 
calculus is founded is this: 
If the laws which regulate the combinations of symbols of operation be the same 
as those which regulate the combinations of symbols of quantity, then all forms which 
would be equivalent relative to the latter, must also be equivalent relative to the former. 
The laws to which symbols of quantity are subject, may be briefly classed un- 
der the seven following heads. 
1. Their affections by numbers, or numerical quantities, are the same as if 
they themselves were numbers, or numerical quantities. 
2. The law of signs. 
3. The order of simple operations is indifferent. 
4. The order of combined operations is indifferent. 
5. Combined operations may be distributed. 
. and 7. The laws of indices. 
Hence, if d, @, ~ are any symbols of operation, subject to these laws (a and 
6 being numerical quantities) : 
1. a+b) p=aptbhp=agPt Hd; &e. 
2. (at) (b= VW=ab=aytig-OGy; &e. 
3. ptdadto 
4. v=o 
5. d(p+y=dpidy 
Grd ad =o" 
on) 
itt. (a*) =q*’ 
results which would be equivalent were d, », » numerical quantities, are equiva- 
lent when they are operations. For example, 

n n n—1 es n—-2 2 
(d+) =d +nd — = gp + ke. 
The symbols of differentiation = = and of difference A,, A, satisfy these con- 
ditions. 
It must be observed, in applying the principle which I have laid down, that 
it is inapplicable, unless it hold with respect to every symbol which enters into the 
operation. It will evidently apply to the ordinary symbols d and A as combined 
with each other, and to the symbols z, y as combined with each other; but it will 
not apply to the symbols d and x as combined with each other, because the fourth 
law is violated by their combination: For example, 
ee Ae tee 
dAv?=Adzx?:; 
