[ 389 ] 



XLVIII. On certain Researches o/Euler. By James Cockle, 

 M.A., F.R.A.S., F.C.P.S., Ban-ister-at-Law*. 



The formulfe employed by Euler at p. 97 (art. 45) of his paper 

 " De Reductione/' &c. in vol. ix. of the Petersburgh Novi 

 Comment a7iif, will be found to give the results 



PQ = (^^ V/t3/, RS =fk^, 



so that, substituting, as we may without loss of generality, 1 for 

 V in art. 36 (p. 93), we are led to 



But (p. 97, art. 44) A = 5(^ + — ), therefore 



and, A being either finite or zero, g and k, PQ and RS, %^ and 



®S vanish, if at all, simultaneously. 



It follows that Eulei-'s later form (arts. 44 to 46), although a 

 generalization of his earlier one (arts. 39 to 42), does not include 

 De Moivre's, nor, a fortiori, the result at which I have arrived J, 

 viz. that 



f-5¥f-5Q,y^-5Ry + ^ = 



admits of finite algebraic solution when the symmetric product 



P(E2-7Q2S)(QSE + Q* + S3) + (11P4S + 13P2SHS3)2 



+ (QSE + Q4 + S3)2+((2Q)8-(5P)(2Q)(3S)}Q(PS)2 



+ {(PQ)^ + PS2+(2Q)2S}(PE)2 + 137(PS-Q2)P4Q2S 



+ 11P3{11P3Q-(2Q)3}SE+{7(5P2S)2 + P2(3S)3}QE 



+ 17P3{Q4 + (3P)(2Q)P2Q-(11P3)(3S)P}Q2-S6+(11P4Q)2P 



PQE{(5P)(2Q^)P3 + (8P)(2Q^)2} + {(5P)(2Q)(3S)2}FQ, 

 or 7r-T-5''*, vanishes. 



It should be observed that tt is an element, not only of a new 

 criterion of solvability, but of the algebraic roots, if indeed any 

 exist, of a general quintic. 



* Communicated by the Author. 



t Pro annis 17(i2 et 1763, Petropoli 17G4, pp. 70-98. When either g, 

 k, m, n, or r vanish, Euler'.s T (at. ji. 97) becomes a square. 



X Phil. Mag. May 1H57, j). 354 ; Quarterly Journal of Pure and Applied 

 Mathematics, May 1857, ]). 144 ; Ladv's and Gentletnan's Diary for 1858, 

 p. 77. 



