OF THE ROOTS OF ALGEBRAIC EQUATIONS. 351 



with the condition, 



0= [^^v — + §y—'^ Q (21). 



d 



dx ^ dy 

 And it will easily appear that 



d'"P / rf \ . . rf°' -P / . d\ , d'"P I d\ 



- — = cos y -— /"(jy), -; r-r = - sin y — ) f" (a;), = - cos « — /'" (x)... 



dw" \ " dxj ■' ^ " dx^-'dy \ ^ daj •' ^ " dx"'-'df \ ^ dw) ■' ^^ 



- — = sin 2/ -;- / Wj -. tt- = cos v — f U), ^^^ — = - sin w — f (w)... 



dx"" \ ^ dxj •' ^ ' dx^-'dy \ •' dxl ■' ^ ' dx"'-'dy^ \ ^ dxj ■' ^' 



Therefore calling the values assumed by sin iy —J /'" (,r) and cos iy — ) /'» (*>) at the point 

 in question A and B respectively, equations (20) (21) may be written, 



1.2 ml''^ = Blx' - inA^af-' Sy- "^ ^'" ~ ^^ BSx"'-' Sy' + &c. 



= JSx'' + mBSx'^-'Sy - "iilJlA JSaf-'W - &c. 

 ' 1.2 " 



Let ix = Ss cos (p, Sy = Ss sin cp, 



d'"« „ / m (m - I) , , . „ \ . , , . 



1.2 ?»n^-W = B Icos"^ ^ cos"'^ (p mn^(p + I - J (m cos'""' <p sm (p - ...) 



(p sin^ (p + J + B {m cos'""' <p sin (p - ...) 



or 1.2 . 



6 s 



= J cos m(p + B sin mtp, 

 and lastly these expressions may be put under the form 



1.2 »« TT« = C COS {m(p + a) (22), 



iis 



= sin(TO^ + a) (23). 



The last equation gives us 



m(p + a = kTT, 



where /,; may have any one of the values 0, 1, 2...(to — 1); hence there will be m branches; 

 also the sign of ^'"z depends upon that of cos kvr, or of (- 1)*, and will therefore be alternately 

 [Kjsitive and negative ; hence the to branches will be curved alternately in opposite senses. 



Hence, therefore, if values of x and y can be found which will make Sz = and Q = 0, there 

 will not be a maximum or minimum point properly speaking, but a multiple point in which 

 two or more branches of the curve meet, and these branches being, as has been proved, curved 

 in opposite senses, there cannot be an absolute maximum or minimum, that is, a maximum for every 

 branch or a minimum for every branch. 



11. This proposition completes the theory of the roots of the equation /(«) =0; for it 

 has been shewn that the curve of double curvature corresponding to the equation x = f(x) admit.s 

 of DO maxima or minima, and that it consists of n branches going off alternately to positive and 

 negative infinity, hence the plane oi xy or any plane parallel to it must necessarily cut the curve in 

 n points, and the distances of these n points from the origin will be the n roots of tlie equation. 



