110 MR. JAMES cockle: SUPPLEMENTARY 



§36. 



Jjet u'=pu + u^, a,^=f{i^') and bn=f^{i'"'), 

 then d{u') = 6{pu + u^) 



= {aip + bj) {a.2p + 63) {a.p + b^) [a^p + b^ 

 = Ay + Bj9'' + Cp^ + D/? + E, say, 

 and A.=^aya^aoa^^=6{ii), E = b^b2b^b^ = 6{u^}, 

 B = a^ai[arj)^ + bcfi,^ + a2a^{a]b^ + b-^a^ 

 =Pt{u)J - pt{u)1= - 2p^r{u)Xv^{tv + z) 



in virtue of (22) and of tlie relation p^=3- 



Again, C = a^aibj)^ + a^a^^b^ + {a^^ + b^a^ {aj)^ + b^a^ 



=Pt[u) [bj)^ - bj)^ + IJ = /9t(m) (6363 - 61^4) + Oi^w^) 

 = 6{u^) - st{u) {2t{u^) - ^v^w^ ; and 

 D = bj)i{aj}^ + b^a.^ + b^b^ia^b^ + biU^ 

 = b,bj + bM = sr{u''){2T{u') - Hv'w^. 

 These results furnish us with the development of d{u) and 

 the only observations* which it seems necessary to make 

 upon them are that the formulae of § 34 give 



and that from 



6263= (r + i^)I!'v^{w^ + z^) + {i + i*)Xv^{x^ + y^) 

 = - pr{u^) + (i + i*)5'vW 

 we pass at once to 



Ms - b^b^ = - p{2t{v^) - 5't'W}. 



§ 37- 

 I now proceed to other points which appear to me to 

 place in a clearer light the relation in which these 

 researches stand to the theory of quintics. Let a, b, c 

 and d be any four symbols, and X a rational function of 

 them. The most general substitution that can be applied 

 to X is one consisting of three successive binary inter- 

 changes, and either side of the equivalence 



* See also my Observations, &c.j Phil, Mag, May 1859. The transformed 

 equation is there exhibited. 



