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VI. Addition to the Memoir on Tschirxhausen's Transformation. 

 By Akthur Caylet, F.R.S. 



Received October 24, — Eead December 7, 1866. 



In the memoir " On Tschienhausen's Transformation," Philosophical Transactions, 



vol. clii. (1862) pp. 561-568, I considered the case of a quartic equation: viz. it was 



shown that the equation 



{a, b, c, d, ejx, 1)^=0 

 is, by the substitution 



y={ax+b)E-\-{aa^ + ihx+Zc)C^{aoi^ + ibx'-\-%cx+2>d)T), 



transformed into 



(1, 0, c, J3, ei^^, 1)^=0, 



where (C, J3, (25) have certain given values. It was further remarked that (C, IB, C!5) 

 were expressible in terms of U', H', $', invariants of the two forms {a, b, c, d, e^JJ^, Y)*, 

 (B, C, DXY, — X)% of I, J, the invariants of the first, and of 0', =BD-C^ the inva- 

 riant of the second of these two forms, — viz. that we have 



C=6H'-2I0', 



e=iu'^-3H'^+r0'^+i2j'0'u'+2r0'H'. 



And by means of these I obtained an expression for the quadrinvariant of the form 



(1, 0, c, B, e:iy, 1)^ 



viz. this was found to be 



=IU"'+|P0'=' + 12J0'U'. 



But I did not obtain an expression for the cubinvariant of the same function : such 

 expression, it was remarked, would contain the square of the invariant 4>' ; it was pro- 

 bable that there existed an identical equation, 



JU'^-IU'^H'+4H"+M0'= - $^ 



which would serve to express <l>'^ in terms of the other invariants ; but, assuming that 

 such an equation existed, the form of the factor M remained to be ascertained ; and 

 until this was done, the expression for the cubinvariant could not be obtained in its 

 most simple form.' I have recently verified the existence of the identical equation just 

 referred to, and have obtained the expression for the factor M ; and with the assistance 

 of this identical equation I have obtained the expression for the cubinvariant of the form 



(1, 0, c, ©, eiy, l)^ 



MDCCCLXVI. P 



