358] ADDITION TO THE MEMOIR ON TSCHIRNHAUSEN’s TRANSFORMATION. 451 
be respectively of the order n — 1, =3, it follows that the equation in y obtained by 
the elimination of x from the equations 
(a, b, c, d, e§x, 1) 4 = 0, 
(a, /3, 7, 8\x, l) 3 
y (o', /3', 7. «'$*, ir 
is a mere linear transformation of the equation A A + BH = 0, where A, B are 
functions (not as yet calculated) of (a, b, c, d, e, a, /3, 7, 8, a!, /3', 7', 3'). 
Article Nos. 1, 2, 3. Investigation of the identical equation 
JU' S - IU' 2 H' + 4AP 3 + if®' = - 
1. It is only necessary to show that we have such an equation, if being an 
invariant, in the particular case a = e = l, b = d = 0, c = 6, that is for the quartic 
function (1, 0, 6, 0, l) 4 ; for, this being so, the equation will be true in general. 
Writing the equation in the form 
- if®' = U' 2 (JIT - IH') + 4ff' 3 + <i>' 2 , 
and observing that we have 
U' = (A 2 + D 2 ) + 26 BD + 4 6C 2 , 
R = 6 (A 2 + A 2 ) + (1 + 6 2 ) BD - 46 2 G 2 , 
©' = BD - G 2 , 
<*>' = (1 - 96 2 ) G (B 2 — D 2 ), 
I = 1 + 36 2 , 
J =6-6 s , 
and thence 
JU' — IH' = — 4<9 3 (A 2 + A 2 ) + (- 1 - 26 2 - 56> 4 ) BD + (8<9 2 + 80 4 ) C 2 , 
the equation becomes 
-(BD-C 2 )M = 
{- 46 3 (A 2 + D 2 ) + (- 1 - 26 2 - 5<9 4 ) BD + (86 2 + 80 4 ) G 2 } x {A 2 + D 2 + 20AA + 46C 2 } 2 
+ 4 {(9 (A 2 + A 2 ; + (1 + 6 2 ) BD - 4<9 2 C 2 } 2 
+ (1-90 2 ) 2 A 2 {(A 2 + A 2 ) 2 - 4A 2 A 2 }. 
57—2