• RESEARCHES IN THE HIGHER ALGEBRA. 109 



where X' corresponds to the cycle vzxyw. Under the 

 cyclical aspect X is singularly efficient as an instrument 

 of calculation^ and a transformation, to which I was con- 

 ducted some time since, enables us to demonstrate with 

 great ease and brevity a proposition asserted in § 28. 

 Mr. Harley's notation (which appears in the same volume 

 of the Memoirs) at once enables us to write 

 /'"(«)/"(«*) = 5'v'""" + iXV^'z" + PXv''Y + i^Xv'^w'' + i^Xv'^W' 



= X'v"'f''{i^) = XV'fii) . 

 Consequently 



I = xv"^{f"{i^) +f"{i) } = xv^'irii) +f^ii') }, 



J = Xv^^{f\^) +f\i') } = Xv^'{f"\P) +f"\i^) }, 

 and I + J and IJ are invariable under interchanges of m 

 and n and powers of i and, inasmuch as the latter is not 

 symmetric, I^ + J^, and, therefore, (S) are functions of r. 



§ 3<5- 

 We may, no doubt, obtain relations connecting epi- 

 metrics of various forms, and indeed I have actually 

 obtained them [vid. Phil. Mag. February 1854 and August 

 18^6). But the following results, obtained with the 

 greatest ease, show the practical superiority of the cyclical 

 process : 



{XvwY= X'vw . X'vw = X' («^W 4- 2V^WZ) 



ov, {r{u)Y=T{u^) + 2X'v^wz', (20) 



X'vX'v^{w + z) — X'v^{w + z) + 2t{u^) + Xv^iwz - xy) 



—X'v^{w + z) + 2r{u^) -{r{u)l^ + ^X'v'-wz, 



or, {t(m)}^ = 5"«^^(««' + 2') + 2t(%^) + 2^'V^2^2'; . . (21) 



consequently, combining (20) and (21), we find 



Xv'{w-\-z)^-t[vF)=o (22) 



These results, which have reference to the trinomial 

 quintic but may readily be generalised, facilitate a trans- 

 formation now to be discussed. 



