122 Review of Mathematical Researches. 



The utmost that we can attempt is, to communicate information concern- 

 ing their contents. 



Those who have but touched the second part of Algebra are aware, that 

 equations above the fourth degree are not solvible in general terms. One, 

 however, who reflects how little use is made of the finite solution of equa- 

 tions of the third and fourth degrees, will see that there is no great result 

 to be anticipated as a direct consequence of solving the higher equations. 

 It certainly is not in itself the field in which one would desire to see su- 

 perior talent spending itself. But when a problem has long baffled ma- 

 thematical ingenuity, then, whether it be solved, or be proved insolvible, 

 either event will ordinarily give us a fuller acquaintance with the whole 

 subject J and the methods by which the question has been settled will often 

 prove fertile in other results. This we feel to be tlie case with Mr. Jerrard's 

 Researches. He has so extended the Calculus of Symmetric Functions, 

 correcting the bad notation of his predecessors, and inventing a most com- 

 prehensive and methodical notation of his own ; that he will as justly be 

 named the author of this new Calculus, as La Grange of the Calculus of 

 Variations. Such an invention is a powerful engine, the effects of which 

 we cannot at first ascertain. 



The use of yi f'2 /3 .... for the sums of the first, second, third .... 

 powers of the roots of a given equation, is familiar to the ordinary books. 

 Newton gave the formula for finding fn, and it is hence easily shown that 

 every rational symmetric function of the roots is a rational function of the 

 coefficients, even though the roots be impossible. The notation /"«»?, 

 fn mr , . . . has also been used to express the union of roots two and two, 

 three and three, . . . . in symmetric functions wherein each term is formed, 

 as .»'"' x"", of" J!""" .t'"'*' .... respectively. But it seems strange, (now that 

 Mr. J. has pointed it out,) that fmni was defined to mean something else 

 than the result obtained in fm n by putting n=^m : viz. the latter was made 

 to be double the former. So ; fm n r, taken when m=.n^=.r ; was made to 

 be (2 X 3) times fm m m. This breach of analogy was an insuperable 

 barrier against the progress of the calculus. Having set this right, Mr. J. 

 is able to generalize and to discover beautiful modes of condensing expres- 

 sions. By separating the symbol f upon the principles advocated by 

 Herschel, he solves with equal simplicity and elegance the general problem 

 of elimination. Given A'=0, A" = 0, two integer functions of .r, to express 

 the relation between the coefficients, independently of x, by an equation of 

 the lowest possible degree. 



I'ut X = jo + y .f + r ,^•2 + + lx"'\ 



X' = P + qx + Rx"- + -h Z/^"/ 



M. J.'s result by the elimination of .r is then 



^ + 1 ^ + 1.2 '^ +....+ x:l...m -" 

 wherein a„ =/(! § -h 2 .B + ... + n Z/)'' 



