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XVI. Algebraical Researches, containing a disquisition on Newton’s Rule for the Discovery 
of Imaginary Roots, and an allied Rule applicable to a particular class of Equations, 
together with a complete invariantive determination of the character of the Roots of 
the General Equation of the fifth Degree , &c. Ry J. J. Sylvester, M.A., F.R.S., 
Correspondent of the Institute of France, Foreign Member of the Royal Society of 
Naples, etc. etc., Professor of Mathematics at the Royal Military Academy , Woolwich. 
Eeceived April 6, — Read April 7 , 1864. 
Turns them to shapes and gives to airy nothing 
A local habitation and a name. 
(1) This memoir in its present form is of the nature of a trilogy ; it is divided into 
three parts, of which each has its action complete within itself, but the same general 
cycle of ideas pervades all three, and weaves them into a sort of complex unity. In 
the first is established the validity of Newton’s rule for finding an inferior limit to the 
number of imaginary roots of algebraical equations as far as the fifth degree inclusive. 
In the second is obtained a rule for assigning a like limit applicable to equations of the 
form ^X{ax-\-b) m =^, m being any positive integer, and the coefficients a, b real. In the 
third are determined the absolute invariantive criteria for fixing unequivocally the 
character of the roots of an equation of the fifth degree, that is to say, for ascertaining 
the exact number of real and imaginary roots which it contains. This last part has 
been added since the original paper was presented to the Society. It has grown- out 
of a foot-note appended to the second, itself an independent offshoot from the first part, 
but may be studied in a great measure independently of what precedes, and constitutes, 
in the author’s opinion, by far the most valuable portion of the memoir, containing as it 
does a complete solution of one of the most interesting and fruitful algebraical questions 
which has ever yet engaged the attention of mathematicians ( 1 ). I propose in a subse- 
quent addition to the memoir to resume and extend some of the investigations which 
incidentally arise in this part. The foot-notes are numbered and lettered for facility of 
reference, and will be found in many instances of equal value with the matter in the 
text, to which they serve as a kind of free running accompaniment and commentary. 
(') I owe my thanks to my eminent friend Professor De Morgan for bringing under my notice, in a marked 
manner, the original question from which all the rest has proceeded. As all roads are said to lead to Rome, so 
I find, in my own case at least, that all algebraical inquiries sooner or later end at that Capitol of Modern 
Algebra over whose shining portal is inscribed “ Theory of Invariants.” 
MDCCCLXIV. 4 I 
