

<cnA.Qi\t2 



> (w- 



-1)A, 2 , 



3(n- 



-1)A,A, 



>2(rc- 



-2)A 2 2 , 



4(»- 



— ^)A 2 A. 4 



>3(rc- 



-3)A 3 2 , 



1 14 Prof. Young on the Demonstration of Newton's Rule for 



— an interest for which I am at a loss to account, since Newton's 

 Rule, whether proved or left unproved, is but of little conse- 

 quence now — I feel naturally anxious that mathematicians in 

 general should judge and decide whether this public condemna- 

 tion of my efforts was hastily pronounced or not. 



I have shown, and very likely others have shown also, that 



k n x n + A n _ 1 B - 1 +A w _ 2 n " 2 + . . . +A 3 ^ + A 2 ^ 2 + A^ + A =0 



being any numerical equation, 

 it will of necessity have one pair 

 of imaginary roots at least, pro- 

 vided any one of the conditions 

 here printed in the margin have 

 place :— 5(rc-3)A 3 A 5 > 4(rc-4)A 4 2 , 



These criteria of imaginary * : 



roots pave the way for the proof . , ' , 1U 2 



of Newton's Rule, which proof ^A„_ 2 A„> \n ■ ijii»_, . 



I now give. 



From the general equation above, the following series of equa- 

 tions are derived in the usual way, namely, 



nA n cc n ~ l + . . . + 4A 4 r s + 3A 3 # 2 + 2A 2 # + A x =0, 

 fi(n-l)A w a? n - 2 + . . . +4 . 5A 5 ^ 3 + 3 . 4A 4 a?« + 2 . 3A 3 # 



+ 2A 2 =0, 



w(^-l)(/i-2)A n ^- 3 +...+4.5.6A 6 ^ 3 + 3.4.5A 5 ^ 



+ 2.3.4A 4 #-f2.3A 3 =(V 

 &c. &c. &c. 



And it is well known that if any of these have imaginary roots, 

 as many, at least, must enter the primitive equation. The 

 same remark applies if we reverse the coefficients of each of 

 these equations, as also if we take the limiting equations derived 

 from them when the coefficients are thus reversed. 



Reversing, then, the coefficients of each of the above equations, 

 commencing with the primitive, it is readily seen that the derived 

 cubic equations will be 



4.5...rcA .r J + 3.4...(rc-l)A r * 2 + 2.3... (n-2)A 9 se 



+ 2.3..>-3)A 3 =0, 

 4.5... (r t -l)A^ 3 + 3 . 4 . . .(n-2)2A 9 a? 8 + 2 . 3 . . . {n-3)3A 3 x 



4-2.3... (rc-4)4A 4 =0, 

 4.5... (rc-2)2A 2 * 3 f 3 . 4 . . . (n— 8)2 . 3A^ 2 + 2 . 3 . . . (w-4) 



3 . 4A 4 # + 2 . 3 . . . (ti~5)4 . 5A 5 =0, 

 &c. &c. &c. 



