678 ME. A. CAYLEY ON TSCHIENHATJSEN’S TEANSEOEIVIATION. 
+( 
be - P, 
fg — ch. 
hf-bg. 
mg— nh. 
fm — bn , 
— fn "bem 
fg — ch, 
ca — g% 
gh— af , 
— gl -ban. 
nh —If , 
gn— cl 
hf — bg. 
gh — af , 
ab— h^ , 
hi —am. 
— hm-bbl , 
If — mg 
mg — nh. 
fm —bn. 
— fii -bem. 
ap — P , 
ph — Im, 
pg-ln 
— gl -ban, 
nh —if , 
gn— cl , 
ph — Im , 
bp — m% 
pf — mn 
hi —am. 
— hm-J-bl , 
If —mg. 
pg -In , 
pf — mn. 
cp— n® 
+(?^+^<r+i/T)"=0, 
the first term whereof, substituting for (a, b, c, f, g, h, 1, m, n, p) their values, is in fact 
equal to the discriminant See. of the quintic (a, 5, c, d, e,f'X^, Y)®. There is no 
loss of generality in putting all but two of the quantities (X, (/j, v, g>, <r, r) equal to zero ; 
in fact this leaves in the formulae a single arbitrary quantity, which is the right number, 
since the ratios B : C : D : E have to satisfy only the two conditions C=0, 2B=0. 
Addition, Nov. 10, 1862. 
I take the opportunity of remarking, with reference to my memoir “ On a New 
Auxiliary Equation in the Theory of Equations of the Fifth Order that I recently 
discovered that the auxiliary equation there considered is in fact due to Jacobi, who, in 
his paper, “ Observatiunculse ad theoriam sequationum pertinentes f ,” under the heading 
“ Observatio de aequatione sexti gradus ad quam aequationes quinti gradus revocari pos- 
sunt,” gives the theory, and observes that the equation is of the form 
<P®-j-<Z2<p*“l-^Z4<P^“f'<Z6= 32<y/ C 
and mentions that the value of is easily found to be (I adapt his notation to the 
denumerate form (a, h, c, d, e,f%v, 1)®=0) 
=40«e— 16M+6c^ 
(this ought, however, to be divided by —2), but that the values of a^, “pauUo am- 
pliores calculos poscunt.” 
The value of the coefiicient in question is correctly obtained (page 270 of my memoir) 
in the form 
—c^ 
-{-2( — 16ae-\-4:bd—c^) 
-\-12ae; 
but the reduced value is given in two places (page 271) as equal to 
— 32 ae, this should be —20 ae, 
%hd, „ +8 id, 
— 3c^ „ — 3c\ 
The last-mentioned correct value was used in obtaining the coefficient for the standard 
form, which coefficient is given correctly, page 274. 
t 
* Philosoplucal Transactions, vol. cli. (1861) pp. 263-276. 
t Ceelie, t. xiii. (1835) pp. 340-352. 
