Mr. J. Cockle on the Method of Symmetric Products. 137 



74. Not to pursue this subject further, let us return to the 

 theory of quintics, and consider the binomial form, in which all 

 the roots are given by 



75. In this case we have 



Y,^0, Y,=5t, Y3=0, Y,=0. 

 7Q. For the form of De Moivre, we have 



2/ = aTi+«'*T4, 



and 



Y, = 5t4, Y2 = 5ti, Y3=0, Y, = Q. 



77. For the first form of Euler, we have* 



y = axi + (f ^T2 or ?/ = axi + a^Tg, 

 and 



Y, = 0, Y2 = 5ti, Y3=5T2orO, Y4=0 or 5r. 



78. For a form obtainable by modifying an assumption of 

 Bezout, we havef 



y = «T,+aV2+a% 

 and 



Y,=0, Y2=5t„ Y3=5t2, Y4=5t3. 



79. This form is the first suggested by the Method of Sym- 

 metric Products. Its attainment would reduce the problem to 

 the following ' Given a quintic with a known homogeneous linear 

 relation (Y = 0) existing among its roots, to find those roots J.^ 



80. Thus far I have traced this symmetric (or epiraetric) 

 method. The discussion of the problem of (79), of the deter- 

 minability of ^{x^x,^^, &c., and of the uses to which the dis- 

 posable members of the series in Q can be put, belong to the 

 general theory of equations of the fifth degree, under which I 

 hope at a future time to reconsider them. In dealing with the 



of the theory of symmetric functions is given by Lagrange (Equations, 



ii. \W). and tables of their values, up to the tenth degree inclusive, will be 

 ounil at p. .S74 of Vandermonde's Memoire sur la resolution des Equations 

 referred to in (53), and published in the year 1/74. Mr. Jerrard's ' Re- 

 searches ' and his recent investigations (Phil. Mag. for May and Supp. for 

 June 185.3) have given a great extension to the theory. 



* Euler, De resolutione, &c. (§§ 39, 40, 41 and 42', pp. 94-6 of vol. ix. 

 of the Pet. New Com. published in 1764). The paper Deformis, &c. was 

 published in 1738. 



t Bv making Bezout's a (Pax-. Mhi. for 1765, pp. 543, 544) vanish wc 

 should' have tlie form of (78). For each of the forms of {lb), {li\), {77) 

 and (78), Tr/yj) vanishes. 



X On til is part of the subject see my paper On Equations of the Fifth 

 Degree at pp. 84-(; of the ' Diary ' for 1848. 



Phil. May. S. 4. Vol. 7. No. 43. Feb. 1854. L 



