T A Y 



cular to a line whose equation is y f y -? *- (x f x), the equation 

 to the normal will be y" y = -- -, (JF" JT). fS<? i/ae, Art. 4J 



Ex. In the parabola, equation to the tangent is y' y --- (.V JT); 

 and that to the normal y" y -j (x" A'). 

 For tangents to Spirals see Spiral. 



TAYLOR'S Theorem. (Higman.) 



If x and y be the coordinates to any point of a curve, and if, when x 

 becomes x -f 7i, y becomes y' ; then will 



dy . dz v 7i% fh y 7i3 



y'=y + -^ K + a-3 15 + s^iAs** 8 - 



Cor. 1. If when x becomes x 7z, y becomes y then will 



Cor. 2. 



Cor. 3. The above theorem may be expressed in general terms thus : 

 The variable of a function being supposed to consist of two parts x and h t 

 to develope the function in a scries of powers of one of the parts h. 



Maclauriri's Theorem. 



To expand a function in a series of ascending- integral and positive 

 powers of the variable. 



Let u any function of x t then u - () -f (-^ ~) *' + (|yO 

 ^8 xc?3 u \ .rs , / > / ^ u \ / fh u \ 



1T2 + U^a) u7i + &c * whcre (w) ' ("ST> C^)' &c - denot9 



the values of u, -^-^, ^-jg &c. when .r = 0. 



This theorem is only a particular case of Taylor's, for take jc = ia 

 Taylor's series, and we have 



which is the same as the theorem above, if for h we write jr. 



