182 



Prof. Sylvester — Discovery of Imaginary Roots. [April 7, 



This theorem was published by the author, but without proof, in the 

 * Comptes Rendus ' for the month of March in this year. 



But the method of demonstration now supplied is deserving of particular 

 attention in itself ; for it brings to light a new order of purely tactical con- 

 siderations, and establishes a previously unsuspected kind of, so to say, 

 algebraical polarity. The proof essentially depends upon the character of 

 every superlinear form being associated with, and capable of definition by 

 means of a pencil of rays, which may be called the type pencil, subject to 

 a species of circulation of a different nature according as the degree of the 

 form is even or odd, which he describes by the terms ''per-rotatory" in the 

 one case, and " trans-rotatory " in the other ; so that the types themselves 

 may be conveniently distinguished by the names "per-rotatory" and "trans- 

 rotatory." By per-rotatory circulation is to be understood that species in 

 which, commencing with any element of the type, passage is made from it 

 to the next, from that to the one following, from the last but one to the last, 

 from the last to the first, and so on, until the final passage is to the element 

 commenced with from the one immediately preceding. By trans-rotatory 

 circulation, on the other hand, is understood that species in which, com- 

 mencing with any element and proceeding in the same manner as before 

 to the end element, passage is made from that, not to the end element 

 itself, but to its polar opposite, from that to the polar opposite of the next, 

 and so on, until the final passage is made to the polar opposite of the ele- 

 ment commenced with, from the polar opposite of its immediate ante- 

 cedent. The number of changes of sign in effecting such passages, whether 

 in a per-rotatory or a trans-rotatory type, is independent of the place of 

 the element with which the circulation is made to commence, and may be 

 termed the variation-index of the type, which is always an even number for 

 per-rotatory, and an odd number for trans-rotatory types. A theorem is 

 given whereby a relation is established between the variation-index of a 

 per-rotatory or trans-rotatory and that of a certain trans-rotatory or per-rota- 

 tory type capable of being derived from them respectively; and this purely 

 tactical theorem, combined with the algebraical one, that the form f(x, y) 

 cannot have fewer imaginary factors than any linear combination of 



leads by successive steps of induction to the theorem in question, 



but under a more general form, which serves to show intuitively that the 

 limit to the number of real roots of a superlinear equation which the 

 theorem furnishes must be independent of any homographic transfor- 

 mation operated upon the form. The author believes that, whilst it is 

 highly desirable that a simple and general method should be discovered for 

 the proof of N;iwLon's mle as applicable to equations of any degree, and 

 that the strenuous efforts of the cultivators of the New Algebra should be 

 directed to the attainment of this object, his labours in establishing a 

 proof applicable as far as equations of the 5th degree inclusive will not 



