20 JOURNAL OF THE 



Let US take as another illustration the common parabola 

 y' = 2px. 

 When X increases to x + lx^ y changes to y -f- i^y 



. • . yP- -f- 2y ^y + {^yf = 2/ {x + ax). 

 Subtracting the first equation from the second 

 2y Ay + {^yf = 2p A.r 

 a;/ 2p 



A.l' 2y + A_y 

 As AT approaches zero, ajk tends in the same time towards 

 zero, a limit which neither can attain however. The right 



2p 

 member similarly approaches indefinitely the constant — 



2y 

 without ever being able to attain it, which is therefore its 

 limit by the definition. Hence slope of tangent at point 



(.r, ;/) is, 



A>' dy p ^ p 



lini. — = — =z-z= — -yA — • 

 Ao; dx y ^ 2x 



From this equation we see that the slope of the tangent 

 varies from plus or minus infinity for .t = o to plus or 

 minus zero for .r =: go. 



^y o dy 

 As, lim. — r^ - ^ — , 

 A.r o dx 

 the dy and dx would appear to replace the zeros in the 



o 

 singular form -, which gave rise to Bishop Berkeley's wit- 



o 

 ticism that the dy and dx were "the ghosts of the departed 

 quantities a^ and A.r." As we have defined them above, 

 dy and dx are finite quantities, of the same nature as y and 

 x, whose ratio is always equal to the derivative, this ratio 

 being variable when the derivative is variable. As y ^=/{x) 

 can always be represented by a locus (since for assumed 

 values of .r we can compute and lay off the corresponding 



