Mr. J. Cockle on the Method of Symmetric Products. 131 



45. Let 



^i-!/\ + ^l/i, Y,i = '2^1 + 2/2. 

 and, consequeutly, 



V+Yi = 2.Y=(r+l)(yi + y,) = (.^ + l)S.2/; 

 then, if we make 



^ + 1 = 0, 



we have an equation analogous to that which occurs in the higher 

 degrees, and the resvJt 



shows that Yj may be obtained by the ex*traction of a square 

 root, and, S . y being known, the quadratic is solved. 



46. It may be well, before addressing ourselves to equations 

 of the fifth degree, to illustrate the Method of Symmetric Pro- 

 ducts by an example in which U„ does not occur*. For this 

 purpose let us take the cubic 



aP + ax^ + bx + c=0. 



47. Guided by (4) we assume 



yi={^i-^)-\ y2=(^2-l)"S y3=(*3-?)~S 



and consequently 



48. The conditions of symmetry are 



'f2(y3)=YiY2=S.y^+ES.y,y2, 

 and the expressions («„ «,), (/?,, ^^) each involve the roots of 



Z^-^2 + l=0 {k) 



49. From the identity 



* Systems consisting of two, three, four, . , p single equations maj' be 

 respectively called dual, tcrnal, quaternal, . ,^7-al systems. That which the 

 diicussioii of quintics has presenteil to us for solution is septuagintal and, 

 in the laiigua>^c of my ' Analysis,' sexdecimary. We h.avc only been able 

 to solve <!') or the ni nbers of this 70-al IfJ-o'ry 4-ic system. The refrac- 

 torj' ((luations have given rise to the function U,, or 2'. For my nomen- 

 clature iti' order I may cite the authority of Garnicr {Ancd. AUj. p. 121). 



t I'Vom this, by excluding nugatory assumptions, we might obtain 

 </.'(/3)=0'(^==)={(^'O)}=-2, or </)'(/3)=2 or -1. 

 The two relations between the fimctions <f>' which are cxprossed in (42) 

 give rise re«])ectivcly to the two eepiations which I have, as Question I. 

 (iHGyj of the ' Diary ' for 1H54, proposed for simultaneous solution. 



