stance
zontal
>er of
n the
j 3 , and
I 2 + 4 2 ;
it the
+ c 2 ) 2 ;
705]
if; for shortness,
We thence find
V = — a 4 — b 4, — c 4 + 2 b 2 c 2 + 2 c 2 « 2 + 2 a 2 b 2 .
V (x 2 — a 2 ) = a 2 (— a? + b 2 + c 2 ) 2 ,
V (x 2 — iß) = iß (a 2 — iß + c 2 ) 2 , V (¿c 2 — c 2 ) = c 2 (a 2 + iß — c 2 ) 2 ,
and therefore also
V 2 a 2 x 2 (x 2 — a 2 ) = 4a 6 b 2 c 2 (— a 2 + b 2 + c 2 ) 2 , &c.
Or, assuming the sign of the square roots,
V ax (x 2 — a 2 )* = 2abc (— a 4 + a 2 6 2 + a 2 c 2 ), V 6« (¿r 2 — 5 2 )* = 2a6c (+ a 2 6 2 — b 4 + b 2 c 2 ),
V C5C (¿r 2 — c 2 )^ = 2a6c (a 2 c 2 + b 2 c 2 — c 4 ),
whence, adding, the whole divides by V and we have
ax (x 2 — a 2 )^ + (x 2 — b 2 )% + cx (x 2 — c 2 )^ = 2a6c,
the second equation. Observe that the second equation rationalised gives an equation
of the form (x 2 , l) 4 = 0; the foregoing value x 2 — 4a 2 6 2 c 2 /A is thus one of the four
values of x 2 .
[Yol. XLVIL, 1887, p. 141.]
5271. (Proposed by Professor Cayley.)-—If co be an imaginary cube root of
unity, show that, if
then
y =
dy
(ft) — ft) 2 ) x + 0) 2 X 3
1 - ft) 2 (ft) - ft) 2 ) x 2 ’
(ft) — co 2 ) dx
(1 - y 2 )% (1 + ft)?/ 2 )* (1 - x 2 )i ( 1 + cox 2 )*
and explain the general theory.
[Yol. L., 1889, p. 189.]
3105. (Proposed by Professor Cayley.)—The following singular problem of literal
partitions arises out of the geometrical theory given in Professor Cremona’s Memoir,
“ Sulle trasformazioni geometriche delle figure piane,” Meni, di Bologna, torn. v. (1865).
It is best explained by an example:—A number is made up in any manner with the
parts 2, 5, 8, 11, &c., viz. the parts are always the positive integers = 2 (mod. 3);
for instance, 27 = 1.11+8.2. Forming, then, the product of 27 factors a 11 (bcdefghi) 2 ,
this may be partitioned on the same type 1.11 + 8.2 as follows,
a 3 bcdefghi, ab, ac, ad, ae, af, ag, ah, ai.
(Observe that the partitionment is to be symmetrical as regards the letters which
have a common index.) But, to take another example,
37 =0.11+ 3.8 + 1.5 + 4.2 = 1.11+ 0.8 + 4.5 + 3.2.
77—2
