C. Y. 
75 
[383 
383] PROBLEMS AND SOLUTIONS. 593 
on the 
be the 
he sibi- 
)nj ugate 
Wangles 
tie sides 
for the 
These two results may also be written, 
Disct. (a, b, c, d, e, f\cc, y) 5 = Disct. (/, e, d, c, b, a][x, yf, 
Disct. (a, b, c, d, e, f\x, yf = Disct. {a, boo 4 , coy 3 , day-, ew, f^x, yf; 
that is, the discriminant of (a, b, c, d, e, f\x, yf is not altered by taking the coefficients in 
a reverse order, or by multiplying the several coefficients by the powers go 3 , to 4 , to 3 , to 2 , to, of an 
imaginary fifth root of unity. Applying these theorems to the form (a, b\,cX 2 , cy 2 , by,a\x, yf, 
3 given 
the discriminant is not altered by changing the coefficients into (a, by,, cy 2 , cA 2 , b\, a) ; 
that is, by the interchange of A and y; nor by changing the coefficients into 
hich lie 
adically 
system 
system 
s. The 
, 7' are 
A with 
j point 
and 77' 
ne Aa; 
(a, boy 4 \, cto 3 A 2 , coy~y~, boyy, a), or [a, b (Ato 4 ), c (Ato 4 ) 2 , c (/ago) 2 , b (ya>), a]; 
that is, the discriminant is not altered by the change of A, y into Ago 4 , yay respectively. 
The discriminant is therefore a rational and integral function, symmetrical in regard to 
A, y, and which is not altered by the change of A, y into Ago 4 , yoy respectively. In virtue 
of the second property the discriminant is a rational integral function of (Ay, A 5 , y 5 ), 
and then in virtue of the first property it is a rational integral function of (A/a, A 5 y 5 , A 5 + y 5 ), 
that is, of Ay, A 5 + y 5 . For the general form (a, b, c, d, e, fffx, yf, if a term of the 
discriminant be a a ¥c^d & e^, then we have a + /3+ r y + 8 + e+ (]y = 8, 5a + 4/3 + 37+2S+e=20 ; 
hence attending only to the indices a, /3, 7 we have 5a + 4/3 + 37 > 20, and therefore 
d fortiori 3/3 + 37 > 20, so that /3 + 7 is =6 at most. It follows that for the form 
(a, b\, cA 2 , cy\ by, afx, yf, the sum of the indices of b\, cA 2 is =6 at most, and 
therefore, even if the index of cA 2 is = 6, the index of A will be only =12, that is, the 
discriminant contains no power of A higher than A 12 : hence considered as a function of 
Ay, A s + y 5 , the highest power of A 5 + f is (A 5 + y 5 f; which completes the theorem. 
of the 
[Yol. hi. p. 90.] 
1687. (Proposed by Professor Cayley.)—To describe a spherical triangle such that 
the angles thereof and of the polar triangle lie on a spherical conic. 
On the sphere, the locus of a point such that the perpendiculars from it upon 
the sides of a given spherical triangle have their feet on a line (great circle), is in 
general a spherical cubic ; if however the triangle be such as is mentioned in the 
above Problem, then the locus breaks up into a line (great circle) and into the conic 
through the angles of the given and polar triangles. 
yf■ 
[Vol. m. pp. 92—96.] 
», 0, 1), 
1690. (Proposed by W. A. Whitworth, M.A.)—If ABC be the triangle formed by 
the three diagonals aa', bb', cc' of a complete quadrilateral aa'bb'cc', then a conic can 
be found having double contact in the chord aa' with the critical conic of the quadri 
lateral bb'cc', double contact in the chord bb' with the critical conic of the quadrilateral 
cc'aa', and double contact in the chord cc' with the critical conic of the quadrilateral 
aa'bb'.