OF THE ROOTS OF ALGEBRAIC EQUATIONS. 



349 



dP 



Now it will be easily seen, that if — represent the partial differential coefficients of P 



dx 



with respect to <r, then 

 in like manner, 





dQ I d\ ^r s 



dP _ dQ 



dx dy 



.(12); 



and similarly it may be shewn* that 



dP 

 dy 



dQ 



dw 



.(13). 



Ill order that Sx may vanish when <v and y vary, we must have 



dP. dP^ 



dtV dy 



-— dx + —- hil = 0. 

 dx dy ' 



Multiplying these equations by -— and , and adding, we have, observing the relations 



(12) (13), 



ldP\- 



Hence also. 



dP\' [dQY 

 \l-y) ^[d^] ='-^ 



dP dQ 



— =0—^ = 0. 



dy dy 



dP dQ 



_- = -— = 0. 



dx dx 



If the values of x and y which satisfy these equations also satisfy the equation Q = 0, this 

 will indicate a singular point in the curve, and we must determine the nature of this point ; to do 

 this we have for the increment of x, supposing the terms of the first order to vanish, 



JS p J2 p J2 p 



2^^ = ii:^^^+^2^^^y + ^^v 04); 



dar doody dy^ 



dP 

 (there is no term involving S^y because its coefficient would be -— which in this case vanishes); 



dy 



■ The rootn of ihe equation f{x) = U may be considered as 

 detennincd by the interseetionn of the curves P~(i and Q=0. 

 These curves have the property of intersecting each other at 

 right angles ; for the equations of the tangents to the two curves at 

 a common point (x, y) are 



dP (IQ . dP dQ 

 which in virtue of the relations j — = -r- a"" -r- =-Tn; "" 



present two lines perpendicular to each other. 



dx 



