358] 
449 
358. 
ADDITION TO THE MEMOIR ON TSCHIRNHAUSEN’S 
TRANSFORMATION. 
[From the Philosophical Transactions of the Royal Society of London, vol. CLVI. (for the 
year 1866), pp. 97—100. Received October 24,—Read December 7, 1866.] 
In the memoir “ On Tschirnhausen’s Transformation,” Philosophical Transactions, 
vol. clii. (1862), pp. 561—568, [275], I considered the case of a quartic equation: viz. 
it was shown that the equation 
(a, b, c, d, e\x, l) 4 = 0 
is, by the substitution 
у = {ax + b)B+ (ax 2 + Ybx + Зс) C + (ax 3 + 4bx 2 + 6cx + 3d) D, 
transformed into 
(1, о, e, s>, i)* = o 
where (G>, 2), (§) have certain given values. It was further remarked that ((£, 2), ($) 
were expressible in terms of U', H', Ф', invariants of the two forms {a, b, c, d, efX, F) 4 , 
(Д G, D\Y,—Xf, of I, J, the invariants of the first, and of ©', = BD—C 2 , the 
invariant of the second of these two forms, viz. that we have 
6 = 6#' - 2/©', 
2) = 4Ф', 
(S =IU' 3 - 3H' 2 + 7 2 ®' 2 + 12J'WXT + 2t'&H'; 
and by means of these I obtained an expression for the quadrinvariant of the form 
(1, o, G, $, g% l) 4 ; 
viz. this was found to be 
C. Y. 
= I U' 2 + f/ 2 ®' 2 + 12</0 / IT. 
57