[ 262 ] 



XXXVIII. Oil a Pvopaty possessed in conunoii by the Primi- 

 tives and Derivatives of the Product q/two Monome Functions. 

 By Mr. Edward Sang*. 



nj^^HE idea expounded by Lagrange, of regarding the suc- 

 -^ cessivc differential coefficients ota function of one variable, 

 as similar functions derived from each other, according to a 

 given law, materially changes the aspect of the fluxional cal- 

 culus. His confined notation, however, which is almost a 

 return to the inconvenient method of Newton, prevents the 

 advantages of his system from being generally appreciated, and 

 divests it of that perspicuity which ought always to be the 

 characteristic of algebraic notation. The elegance of his me- 

 thods, as well as their great power, interested me in procuring 

 a more convenient notation, — one which might enable me to 

 pass from one primary to another without being distracted by 

 the confusion which arises from his varied accents. 



In denoting a derived function it is evidently necessary to 

 indicate both the number of times the derivation has been re- 

 peated, and the quantity which, in these operations, has been 

 regarded as the independent or primary variable. Of the three 

 notations which have been employed, that of Leibnitz alone 

 serves both of these purposes ; but from its complexity, as well 

 as from its fractional appearance, it is any thing but convenient 

 in complex operations. 



The particular theorem, which it is my intention in this 

 paper to expound, I have been unable to express by the no- 

 tation either of Lagrange or Leibnitz, without introducing an 

 extension of signification too far-fetched to be admissible ; and 

 am therefore compelled to explain the particular algorithm 

 which conducted me to the result in question. 



If u be a function of z I propose to denote the differential 



coefficient -r^ of Leibnitz or the ^^fonction prime u' " of La- 

 grange by the expression ^u; the second differential coeffi- 

 cient -^ or the Jbnction seconde u'' by ^u, and so on. 



This notation, in the first place, enables us to express in a 

 very clear manner those complex derivatives which are ob- 



. d^ u . 



scurely indicated by such expressions as j^i-jri which from 



analogy one would be very apt to suppose equivalent to 



— ^^— ^ . In fact the quantity intended by the first of these 



expressions, being obtained by deriving three times with x as 



* Communicated by the Author. 



a primary 



