ME.  A.  CAYLEY’S  FIFTH  IMEMOIE  UPON  QUA^TICS. 
447 
Hence,  if  OTj,  Wg,  are  the  roots  of 
(1,  0,  -M,  1)^=0, 
the  expressions  IH — cj-iJU,  IH  — IH  — ro-gJU  are  each  of  them  squares;  write 
= z^ 
so  that,  identically, 
X^+Y^+Z^=0; 
and  consequently  X + <Y,  X— /Y  are  each  of  them  squares.  The  expression 
aX-4-|(3Y+yZ 
will  be  a square  if  only 
which  may  be  seen  by  writing  it  under  the  form 
i(a+;(3)(X-,Y)+i(<.-,/3XX+-Y)-7*VxqW»; 
and  in  particular,  writing  \/ ^ tu^ — ro,,  ra, — for  a,  /3,  y,  the  expression 
)x/lH  roiJU  + (cT3 ^i)\/lH  — — '^3  JU 
is  a square ; and  since  the  product  of  the  different  values  is  a multiple  of  (this  is 
most  readily  perceived  by  observing  that  the  expression  vanishes  for  U = 0),  the  expres- 
sion is  the  square  of  a linear  factor  of  the  quartic. 
131.  To  complete  the  solution:  -us^  are  the  roots  of  the  cubic  equation 
(1,  0,  -pi,  MX^,  1X=0; 
and  hence,  putting  for  shortness, 
P=PI{(-1,  |M,  -pi,  M+^M^XIH,  JUX+x/1-^M(1,  0,  -iM,  MXIH,  JUf, 
Q^=pi{(_l,  pi,  -pi,  M+^jVPXIH,  JUX-yi-^^-M  (1,  0,  -iM,  MXIH,  JVf, 
we  have  (<y  being  an  imaginary  cube  root  of  unity) 
And  if 
then 
Hence, 
1(01  - ^^)  (.173  - 7173)(IH  - = P - Q. 
p;=iM{-i+Vi-iVM) 
— s<“)(3>j — IS,) = P, — Q„. 
multiplying  and  observing  that  (01—01^)^= — 3,  we  find 
~ (a,_a,2)2(^2  — ^3)XIII  — JU)  = (P  — Q)(Po  — Qo), 
and  consequently 
(>».-=».)x/lH-ar.JU=(4.-</)y-(P-Q)(P,-Q,)- 
3 0 
MDCCCLVIII. 
