74 Mr. Jarrett on Algebraic Notation. 



27. The letter D being considered as an abbreviation of the 

 word difference, the symbol D^^.u will denote the w'" difference 

 of u, X being the independent variable, and the increment of x 

 being arbitrary. The case in which Dx, or the difference of x, 

 is unity being of very common occurrence, we may denote by 

 A/ . u, the n* difference taken on that supposition ». 



28. We sometimes need a symbol to denote the new value 

 which a function of any variable assumes on giving to that variable 

 an arbitrary increment. Arbogast has proposed to use the letter 

 E (Etat), and this may be adopted with advantage provided we 

 add the index subscript, to denote the independent variable : 

 thus, -Er-(") will denote that x+Dx has been substituted for x 

 in u ; and E"^ (m) will signify that this operation has been performed 

 n times. 



29. By a slight modification of a notation proposed by Euler 



we may greatly abridge the method of expressing partial diifer- 



d"* + " M d" d" 



ential coefficients. Instead of y-;^ — j—;, he proposed to write — . — . u; 



if we change this last into d"'^d'\j . u we get an expression much more 

 simple. Should the quantity to be differentiated be a power of 

 the independent variable itself, the index subscript may be omitted; 

 thus d.x" may be written for d^.x". 



30. The w'" differential coefficient appears most frequently 



with [M as a divisor ; we may therefore abridge our notation still 



d" u 

 farther by writing- d\. -. u for — ^- . 



- ■' \n 



* Prony, among others, has used tlie index subscript to denote the independent variable; 

 but no notation has hitherto been adopted to distinguish the case in which Dx is arbitrary 

 from that in wliich it is unity. 



