Review of Mathematical Researches. 1 23 



It will be understood that in the last expression, after developing the part 

 beneath y, this symbol is to be applied to each terra separately, always 

 remembering that Q, R, . . . .L, n, are not to be affected with it. Alsoy 

 has reference to the roots of X=0. 



Partially modifying the problem, if X = 0, and y=^(px, (an integer 

 function) we obtain an equation in y ; 



A A A 



«'" — -r' J'"~' + T^ y"'~- — ^c + , „ " = 



"^ 1 '^ 1.2 "^ ^ — 1.2. . .?» 



wherein, if 9 ^ = P ^" + Q xf^ -{■ . . . . + L a.'^ 

 A^ = /(a P + 13 Q + ...+ X L)l^ 

 This result may be obtained immediately from the known relation between 

 the roots and coelBclents of an equation, by help of Mr. .T.'s section. He 

 proceeds farther, to assign a ft . . .\, P Q . . . L such that the equation 

 in y may fulfil particular conditions. Tliis problem could not even be 

 contemplated before it had been shown how to express the equation 

 in y. He first inquires how to reduce a given equation of the m^^ 

 degree, X = 0, to a form in which its second, third, and fourth terms 

 shall have all disappeared. This he solves most satisfactorily ; and when 

 it is known what mathematicians have failed in the attempt, we are war- 

 ranted in adding, with great ability. No one can read his XXII. "'* section 

 intelligently, and not admire his inventiveness. The same mode of treating 

 the subject is carried through his third part, therein he proposes to deter- 

 mine the arbitrary constants that enter (p ,v, so as to make the equation in 

 g solvible, when »j = 5, or when in = 6. In the former case he reduces 



it to De Moivre's solvible form y^ + b y^ + — b'^ y + e = : in the lat- 

 ter, by exterminating the odd powers, he makes the equation virtually of 

 the third degree. All this is likewise satisfactory. But it remains to in- 

 quire whether when y is known, the equation .Y = and (<p ,v — y) = 0, 

 lead to tlic determination of the factors of X, by means of equations of lower 

 diuiensions than X. And here we fail, if {f x — y) is an algebra'c mul- 

 tiple of X. Indeed, since a /3 y . . . X are unlimited in number, and are 

 arbitrarily taken from the series 0, 1, 2, 3 .... ; it seems tiie height of 

 improbability, that in the case of a given function X, this result should 

 always recur. Mr. J., however, has not entered into the disproof: and as 

 far as we have been able to follow the question, it is not to be managed in 

 very few words. In Part IV."" we shall look for it. 



We have said nothing on Mr, J. 's notation for symmetricyrtc/or.?, and on 

 the new symbol 9-. Tiic truth is, that all this was unnecessary for the 

 immediate object of the three parts already published ; and while we ad- 

 mire his aptitude and ingenuity in this matter, we apprehend the book is 

 hereby made less readable. But, (to judge by the specimen in a long 

 note, pp. 56 — 59), when he gives us the rest of his interesting specula- 

 tions on the symmetric functions, we shall not complain that his a is 



