DIFFERENTIAL AND INTEGRAL CALCULUS." 11 



therefore 



dy 



dx a 2 + x* ' 

 (13) y = 



dy 



=a ^ 



dy _ a 



~dx~ x*J(x*-d*}' 



Students often find it difficult to remember when and 

 where to put the a which occurs in some of these differential 

 coefficients. This can always be settled by considering 

 x and a to represent the lengths of certain lines, which 

 in fact they generally do. In all these cases (9)... (13), 

 y being the circular measure of an angle is a pure abstract 



number, whence - . and therefore ~ must be a symbol 



AOJ ' dx J 



represented by an abstract number -=- a number representing 

 the length of some line (where the unit of length may 

 be any whatever), or in other words each result is of 1 

 dimensions in space. Hence, when \J(c? x 2 } is the de- 

 nominator, the numerator will be a number only, or no 

 a is wanted ; but when the denominator is d 2 + x\ or 

 x \l(x* a 2 ), we must have a symbol in the numerator which 

 is of one dimension in space, and the a is required. The 

 proper sign to be affixed can be always supplied by con- 

 sidering whether the function is one which increases or 

 diminishes as x increases. If the former, the sign is of 

 course positive, since Ay, A# must be then of the same 



dii 

 sign, and therefore is positive, if otherwise negative. 



Thus 



d ( . _Jx\} d ( _^x\} 



-5- xsm n- , -=- i tan - n- are positive, 



dx ( WJ J dx ( Wj 



d ( VaA) d ( _Jx\ 

 -y- ^cos )r * j--|cot r ( negative. 

 dx { Wj *dx 1 \aj 



