Prof. Young on Newton's Rule for Imaginary Roots. 291 



in the final triad of the cubic into which it first enters. Should 

 this be the terminal cubic, the examination ends, and nothing 

 can be inferred beyond what was previously known. But if 

 other cubics still remain, and the failure or succession of failures 

 be followed by a fulfilment, this fulfilment must occur in the 

 final triad of the cubic where it first appears. And it must indi- 

 cate a new and independent primary pair. 



This last conclusion I regard as an axiomatic truth — as a prin- 

 ciple of which, though no special proof is given, no special proof 

 is needed by a mind clearly apprehending the antecedent rea- 

 soning. To such a mind the following obvious considerations 

 can scarcely fail to suggest themselves. 



1. The condition (1) is applied exclusively to the triads of the 

 successive cubics, and not to those of the primitive equation at 

 all. Whatever knowledge we gain of the character of the roots 

 of this primitive, we derive from the character of these triads as 

 tested, set after set, by the condition (1), our inferences, how- 

 ever, being always controlled by the fact that each triad consists 

 of three terms of a cubic equation, so that the two triads fur- 

 nished by every such cubic can never imply more than one ima- 

 ginary pair. And however many of the successive triads fulfil 

 the condition (1) of primary pairs, sufficient reason is given why 

 the inference of multiple pairs is forbidden : every final triad 

 transmits its character, in reference to the condition (1), to the 

 leading triad of the next cubic. 



Since a fulfilment after a failure (as also a failure after a fulfil- 

 ment) always takes place in a final triad, it is plain that such a 

 fulfilment is entirely owing to the coefficient last brought down 

 from the primitive, to complete the cubic, being of a certain suitable 

 value. By simply altering the value of this coefficient, we could 

 expel the imaginary pair from the cubic, replacing it not only by a 

 real, but even by an equal pair, all the other coefficients remain- 

 ing untouched. And consequently, by this change of a single 

 coefficient of the primitive equation, a pair of primary roots 

 would disappear. In a cubic not so constituted, that is, in one 

 not consisting of a failure followed by a fulfilment, such conver- 

 sion of imaginary roots into real could never be brought about 

 by any alteration in the coefficient brought down from the pri- 

 mitive to complete that cubic. 



Suppose that from a preceding cubic one primary pair I x in the 

 primitive has been detected, and let the primary pair now indi- 

 cated be denoted by I 2 . This latter primary pair is wholly de- 

 pendent for its existence, as such, upon an advanced coefficient 

 of the primitive equation, of which coefficient the primary pair 

 Ij is wholly independent. It is impossible, therefore, that I x 

 and I 2 can be one and the same pair. And I submit that, to a 



U2 



