349 



or, (4:A±A 2 -A 2 3 )A 0 >A±A l 2 . 



And all the roots will still be imaginary if the sign = replace the sign 

 >, since each of the two component parts of the equation, in the above 

 form, will then be a positive square. 



(12) As implied iri the title of it, one object of the present com- 

 munication is to prove the truth of the rule proposed by Newton, 

 in the Arithmetica Universalis, for determining the number of imaginary 

 roots in an equation, whenever the coefficients of that equation have cer- 

 tain specified relations among themselves — the relations, in fact, which 

 are among those exhibited in the conditions [3], at page 344. In those 

 equations, in which no one of these conditions is satisfied by the co- 

 efficients, Newton's rule is of no avail ; although, notwithstanding the 

 non-fulfilment of any of the conditions adverted to, the equation may 

 have even all its roots imaginary. Yet, restricted as it thus is to the 

 class of roots which we have called primary pairs, when two or more 

 of such pairs enter an equation, the rule will frequently enable us to 

 detect their presence ; while the criteria [3] alone, how many soever 

 of them might be satisfied by the coefficients, could never assure us of 

 the existence of more than a single pair. By the aid of the general 

 theorem at (7), Newton's Rule may be demonstrated as follows: — 



(13) Eeturning to the primitive equation [/], suppose we were to 

 deduce from it the several limiting equations in order : we know that 

 the coefficients A 0 , A x , A 2 , &c, would disappear one after another, the 

 leading coefficient, A n , being the only one that would be retained at 

 the end of the operation. Suppose, now, the order of the coefficients 

 to be reversed, as well in the primitive as in each of these derived equa- 

 tions, and that then the series of limiting equations be deduced from 

 each of these, till we arrive at a limiting equation of the third degree ; 

 then, leaving blanks for whatever numerical factors may have been in- 

 troduced into the coefficients by this process of derivation, the derived 

 cubic equations will be 



Now, for the purpose in hand, we are not interested in knowing what 

 the numerical quantities are which would correctly fill up these blanks ; 

 it is sufficient for this purpose that we know, from the property estab- 

 lished at (7), that, if we were to apply to each of these cubics the cri- 



[ ]A* 3 +[ ]^ 2 + [ 

 [ ]^ 3 + [ ]a 2 x* + [ 

 [ ]a 2 x*+i (aw + I 



] A 2 x + [ ] A 3 = 0 



+ [ }^ 5 = 0...[7/] 



[ ~]A n ^x* + [ ~]A n _ 2 xH[ 



] A n . 1 x'+ [ ] A u = 0. 



