248 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



tioiis, it is, perhaps, necessary to say a few words. The principle on which that 

 calculus is founded is this : 



If the laws ivhich 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 e(ju,ivalent relative to the latter, must also he equivalent relative to the former. 



The laivs 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. 

 (J. and 7. The laws of indices. 



Hence, if d, (p, 4 are any sjonbols of operation, subject to these laws [a and 

 /> being numerical quantities) : 



1. (a =*= b) <p = a <p±b (f> = a (p d=(p b; &c. 



2. (o ± (|)) (6 =i= 4/) = a 6 =F a 4. =h 6 (/) - (^ 4 ; &c. 



3. (^ + ■v|/='v|/ + (^ 



4. (p -^ = -1^ (p 



5. rf((|) + 4/)=(^(^+<^4 



6. (t(i'=<f"' 



7. (effect'"' 



results which would be equivalent were d, (p, 4 numerical quantities, are equiva- 

 lent when they are operations. For example, 



n 11 11 -1 n(n — l) "■-- - 



(d+(p) = d + nd <p + -^ — g-^rf (P + &c. 



The symbols of diiferentiation ^— , -— and of difference Aj,, Ay satisfy these con- 



ditions. 



It must be obsers^ed, 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 x, 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, 



di\x^ = 2, i^dx'-2 

 dii.x' = Adx^: 



