Mr. Jarrett on Algebraic Notation. 75 



31. It is frequently necessary to give a particular value to 

 the independent variable after the difierentiation has been per- 

 formed. This may be denoted by placing the particular value 

 of the independent variable, as an index subscript, to the right 

 of that which denotes the general value of independent variables. 

 Thus d\^^.u signifies that, after taking the w'" differential co- 

 efficient with respect to x, we must put x = a. 



'32. The same notation being extended to integrals, f.,t,u-f,,^u, 

 or, as it may be still more conveniently expressed, (X,(,-X,<.)" 

 will denote the integral of u taken with respect to x, between 

 the limits a and b. In the same manner we shall have 



33. The following examples will best show the application 

 and utility of our notation. 



Ex.1. x-= \ + S^,\ri_.^I^ +(^x-\y^\S^'-*\n-s.~; 



n being a positive integer*, 



x^ — \ 

 For, — -^ = S\jr~\ by division, 



and X"- ^ \ + (x- l) . S'^x"-', 

 by multiplication and transposition. 



Whence, by successive substitution, 



X" = 1 + {x- i).S\{i + (x-i). 5^;-' 

 {H-(x-l).S,;.-'{...{i+(.r-l).5,V.-'^r,-i}...| 



* The first four examples are taken from " The true developement of the Binomial 

 Theorem," by Mr. Swinburne and the Rev. T. Tylecote, of St. John's College; a work 

 which contains the only rigid proof of that theorem that has yet appeared. 



K 2 



