OF THE ROOTS OF ALGEBRAIC EQUATIONS. 353 



17- Hence also we see the truth of a theorem, of which I shall presently make use, namely, 

 that an equation has at least as many imaginary roots as any one of its derivatives; for the equation 

 f(x) = has as many isolated roots as there are imaginary roots in /'(«) = 0, and therefore has at 

 least as many imaginary roots ; f'(v) — has in like manner at least as many imaginary roots as 

 fix) = 0, and so on : whence the truth of the proposition is clear. 



18. If the plane of xy should happen to pass through a real maximum or minimum, which 

 is as we have seen properly speaking a multiple point, there will he several equal roots. The 

 condition of equal roots will he therefore that the plane of xy shall pass through a point for which 

 one or more of the differential coefficients of /(,r) vanish, or which is the same thing, that /(.i?) = 

 and/'(.r) = shall have one or more roots in common ; which as is well known is the test of equal 

 roots. Or we may shew directly that at a point for which there are m equal roots there are m 

 branches curved in opposite senses ; for let w = for simplicity's sake be the root which occurs m 

 times, then 



f{x) = x" \p„ + p,_x!B + + x"-"], 



and the equations to the curve will be 



z = j»„//" cos mQ + , 



= p„ sin mQ + ; 



therefore near the origin, sin niB = 0, 



.•■ mQ = kir where k may =0, l...(m - 1), 

 and ss = p„p"' cos feir 



= (-i)V«i»'"; 



therefore there will be m branches curved in alternately opposite senses. 



19. It will be seen, that a pair of equal real roots in the equation /'(O) implies a pair of 

 imaginary roots in the equation /(.r) = 0, since f"{x) will also vanish for the same value of x 

 as that which makes f'(x) = 0. And generally, if x be even, r equal roots of the equation 

 f'{x) = imply r imaginary roots in the equation f{x) = ; if r be odd, there will be >• + 1 or 

 r-1 imaginary roots according as f(x) and f''*'^{x), which is the first derived function which 

 does not vanish, have the same or different signs. 



This theorem requires no demonstration, as its truth will be seen at once on examination. 

 By means of it I am able to prove the ordinary proposition relative to the number of imaginary 

 roots belonging to an equation defective in any of its terms ; the proof is as follows : 



Suppose, 



f(x) = p„ + p„.iX + + Mx^ + Nx^ + ' + 1 ^ + x", 



where v terms are wanting between the terms Mx'^ and Nx'' + " + ' ; 

 differentiating /u times, we have 



.f (.r) = /a(/"- 1) 2.1 M + (iui + v + l)(ii. + f) ... {v + 2)Nx'^' + 



+ n {n - I) ... (n - fi + 1) a?""" ; 

 differentiating again, 



f''*'(x) = (ji + v + l).(n + v) ... (v + l)Nx" + + w.(w - I) ... (w - iui)x''-''-'. 



Hence the equation 



/''+'(,r) = 



has i; roots equal to 0, and therefore the equation f^x) = has c imaginary roots if v be even, 

 and if v be odd, it has c + l or i; - 1, according as /''(()) and /'"^"'^'(O) have the same or opposite 

 fligns, that is, according as M and N have the same or opposite signs. But, by a theorem cited 

 Vol.. VIII. Paht III. Zz 



