PBOFESSOE CAYLEY’S EIGHTH MEMOIB ON QH ANTICS. 
531 
Article Nos. 286 to 293. — Hermite’s new form of Tschirneausen’s transformation , and 
application thereof to the quintic. 
286. M. Hermite demonstrates the general theorem, that if/j#, y) be a given quantic- 
of the w-th order, and <p(x, y ) any covariant thereof of the order n—2, then considering 
the equation f(x, 1) = 0, and writing 
/>. i) 
(where fl{x, 1) is the derived function of f(x, 1) in regard to xj, then eliminating#, we 
have an equation in z, the coefficients whereof are all of them invariants oi f(x, y). 
287. In particular for the quintic f(x, y) — (a, b, c, d, e,ffx, y)\ if 
^(x,y), <Pix,y), <p 3 (x,y), <p 4 (x, y) 
are any four covariant cubics, writing 
1) +uf^(x, 1) +v<p 3 (x, 1 ) + w<p 4 (x, 1) 
/>, 1 ) 
(viz. the numerator is a covariant cubic involving the indeterminate coefficients t, u, v, w) 
then, in the transformed equation in z, the coefficients are all of them invariants of the 
given quintic. Conducting the investigation by means of a certain canonical form, which 
will be referred in the sequel, he fixes the signification of his four covariant cubics, 
these being respectively covariant cubics of the degrees 3, 5, 7, and 9, defined as follows ; 
viz. starting with the form 
= — 3 No. 16, 
= — 3(A, B, C, D5>, y)\ or (-3A, -B, -C, -3Dfe y)\ suppose, 
and considering also the quadric covariant 
(a, yjx, yf, = No. 14, 
then p,, p 2 , <p 3 , <p 4 are derived from the form 
(A, B, C, BJZ > x-^(3x+2 yy}, Sy+v(2 a x+Py))\ 
viz. we have 
*i(*,y) =-3(A, B, C,D1 x,y)\ 
y) =+3(A, B, C, D%x, y)\-(3x-2yy, 2 ax+fa), 
{<p 3 {x, y)} = - 3(A, B, C, DJ#, y) ( — /3ar. — 2 yy, 2ccx+fiy)\ 
y)} = + 3( A, B, C, -(3x^2 yy, 2ax+(3y)\ 
where {<p 3 (x, y)} and {<p 4 (x, y)} are the functions originally called by him <p 3 [x, y) and 
4 c 2 
y\ -y 2 x yx\ -x 3 
a, b , c , d 
b , c , d , e 
c, d , e , / 
