78 DIFFERENTIAL AND INTEGRAL CALCULUS. 



I should guess no one had ever the patience to work this 

 out according to rule. 



If x, y are connected by an equation, so that each is 

 what is called an implicit function of the other, and 



F(XJ y) = be the equation, the equation for finding - 

 would, according to the general rule for differentiating 



complex functions, be (-T-) + [-3- ] 3^ = 0, the brackets 

 \dx J \ay) ax 



denoting partial differentiation. If this should reduce 

 -j- to the form - , for a pair of values x = a, y = , which 



Ct/tJC 



satisfy the equation F(x, y} = 0, it is best to put x = a + 7{, 

 y = b + k in the equation, and find the value of the limit of 



j directly from the equation. Since k is A?/, and li is A#, 

 this limit will be the value of -^ . The appearance of the 



form - indicates a multiplicity of values ; and if the curve 

 F(x, #) = be drawn, the point will be what is called 

 singular, there being two or more values of -jj- at the point, 



i.e. two or more tangents to the curve. 



Thus, taking an example from Todhunter, if the equa- 

 tion be - 



both of which vanish when aj=l, y=l. Put x=l+x, 

 y = l -f T/, and we have 



{y'* -x' z + 2 (y 1 - x')} x' (x 1 - i) - 2 (y" 2 + x " 2 + 2/) 2 = 0, 



or x (y x) 8j/' 2 + terms of higher dimensions which 

 vanish compared with those retained when ic' = 0, y' = 0. 



