552 
PROBLEMS AND SOLUTIONS. 
[485 
and substituting these values, the result is found to contain the terms 
coefficients which vanish ; viz. the coefficient of the first of these terms is 
+ 16875+24300+ 6561 + 7560 + 18792 - 74088, = 0 ; 
and the coefficient of the second of the two terms is 
- 16875 - 36450 - 19683 - 75168 + 148176, = 0. 
The remaining terms give 
+ 625 
- 5625 - 4050 - 1890 + 9261 
+ 1890 - 18522 
- 18792 + 74088 
+ 9261 
/ 
which is the required result ; a more convenient form of writing it is 
(55296 /, -768 I 2 , — 5544 //, 625 I s + 9261 / 2 $i/ a) 3 = 0. 
IH 4 
= + 625 a 3 ! 3 
= - 2304 alVP 
= - 16632 a-HIJ 
= + 55296 H 3 J 
= + 9261 a 3 / 2 
= 0, 
IP 
a 3 
with 
Remark. If I and / denote as above the two invariants of the form U ={a, b, c, d, e\x, l) 4 , 
and if we now use H to denote the Hessian of the form, viz. 
H = (ac- b\ ^(ad — bc), % (ae + 2bd — 3c 2 ), |(be — cd), ce — d 2 fijx, l) 4 , 
then it appears by the theory of invariants that the equation of the twelfth order 
(55296 /, — 768/ 2 , - 5544//, 625 I 3 + 9261 / 2 #i/, U) 3 = 0, 
is such that each of its roots forms with some three of the roots of the equation 
U = 0 a harmonic progression ; viz. if the three roots are ¡3, <y, 8, then we have 
2 1 1 2/38- (B + 8)y 
~ = Q + s > or x = —^--4— 0 - ; 
x — y x — /3 x — o /3+6 — 2y 
so that the roots of the equation of the twelfth order are the twelve values of the 
last-mentioned function of three roots. 
[Yol. v. pp. 65, 66.] 
On the Problems in regard to a Conic defined by five Conditions of Intersection. 
There has been recently published in the Comptes Rendns (t. lxii. pp. 177—183, 
January, 1866) an extract of a memoir “ Additions to the Theory of Conics,” by 
M. H. G. Zeuthen (of Copenhagen). The extract gives the solutions of fourteen pro 
blems, with a brief indication of the method employed, for obtaining them. Of these 
problems, four relate to intersections at given points, the remaining ten are included