113 MR. JAMES COCKLE : SUPPLEMENTARY 



whereof the coefficients are symmetric functious of a, b, c 

 and d. 



§ 42. 

 The conjugate interchanges now play an important part. 

 To the equation 



where R denotes a rational and symmetric function, apply 

 every possible interchange. The result will be known by 

 examining the effect of the binary changes alone, and we 

 find 



fn=R{(9n, 6'(«^)n} =R{^n, e{^'')} 



^n=R{^(*'0, ^c"')n}=R{^n, ^c^} 

 ^(^'')=R{^n, e{:"'){"')\=^{d{"'), 6'(??)}, 



consequently f cannot receive, by permutation of a, b^ c 

 aud d more than the three values 



which we may call fi, ^o, and ^3 respectively, and which 

 are the roots of a cubic that may be written 



and whereof the coefficients are symmetric in a, b, c and d, 



§ 43- 

 Write 9^ in place of 0, and let 



e,=eci'), e,=d{'-), ^,=^(°^), 



the foregoing discussion shows that if 



^1 = ^1 + 9i, ^2=^3+^6, ^3=^3 + 0^, 



then 



^^^h=^^A2- {OA+OA+OA), or 

 'y=9A+0A+dA-T^„ 



and 7 is symmetric in a, b, c and d. 



