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275] on tschirnhausen’s transformation. 
or A« = — 1. In AF, the part which depends on the variation of A« then is 
— aB + (— 2ax — 4b) G + (— Sax 2 — 8bx — 6c) D, 
and the other part of AV is at once found to be 
+ aB + ( 4a«+66)C + ( 4a« 2 + 126« + 9c) D; 
whence, adding, 
A V = 2 (ax + b)C + ( ax 2 + 4bx + 3c) D, 
and this is precisely equal to 
(Dd c +2Gd B )V; 
so that V is reduced to zero by the operator A — (Dd c + 2Cd B ). 
Similarly, if the second operator is represented by 
V-(2 Cd D + Bd c ), 
then we have 
(a, b, c, c?$«, l) 3 V« + x (b, c, d, e§oc, 1) 3 =0, 
which by means of the equation 
(a, b, c, d, e\x, 1) 4 = 0 
is reduced to Vx = x 2 . Hence in V V the part depending on the variation of x is 
(ax 2 ) B + (2ax 3 + 4bx 2 ) G + (Sax 4, + 8bx? + 6cx 2 ) D, 
and the other part of VF is at once found to be 
(4bx + 3c) B + (4bx 2 + 12c« + 6d) C + (4ba? + 12c« 2 + 12c?« + 3e) 1) ; 
and, adding, the coefficient of D vanishes on account of the quartic equation, and we 
have 
V V= (ax 2 + 4bx + Sc) B + 2 (ax 3 + 4<bx 2 + 6cx + 3d) G, 
which is precisely equal to 
(2Cdi> + Eb c ) V, 
so that V is reduced to zero by the operator 
V-(2 Gd n + Bd c ), 
which completes the demonstration; and the demonstration in the general case is 
precisely similar. 
In the case of the cubic equation we have 
(a, b, c, d\x, 1) 3 = 0, 
y = (ax + b) B + (ax 2 + 36« + 2c) G; 
C. IV. 
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