ME.  A.  CAYLEY’S  FIETH  MEMOIE  UPON  QTJANTICS. 
453 
or  putting  for  shortness 
„ A2  + B2  + C2 
^—2  (B-C)(C-A)(A-B)’ 
we  have 
M=|(A"+B^+C^)A^; 
and  it  is  then  easy  to  deduce 
Z3'j  = A(B  — C) 
nr2=A(C-A) 
273=A(A— B); 
m fact,  these  values  give 
US  j^2  d”  ^ ^2^3 ““IM- 
/y  j ^lT  2 3 ^4, 
and  they  are  consequently  the  roots  of  the  equation  z?®— M(ra'  — 1)=0, 
142.  The  leading  coefficient  of  IH — tViJU  is  then  equal  to  into  the  following 
expression,  viz. 
^A^+B^+e)a->c-5^)-^(A^+B^+e)(B-C), 
which  is  equal  to 
^(A^+B^+e){48a-Vc-5^)~4(B-C)}, 
and  the  term  in  { } is 
8(a[34-a7+a^+|3y+|3S+7^) — 3(c4+j3-l-7-|-B)^ — 4(7  — a)(^  — §)4-4(a — 3)(7  — ^), 
which  is  equal  to 
— 3(S+a— /3— 7)". 
But  IH— z«r,JU  is  a square,  and  it  is  easy  to  complete  the  expression,  and  we  have 
a-<(IH-=>,JU)=-3i4(A=+B'+C’){(8+«-/3-,.,  -ia+liy,  S«((3+y)-(3r(S+«)Iii-,  nfV, 
a-(IH-=,JU)=-3i4(A»+B'+e){(S+(3-y-«,  S/3(r+a)-r»(8+(3)I^,  J/f)”. 
a-(IH-®.JU)=-^(A»+B’+C’){(S+r-a-/3,  -Sr+»/3,  V(«+/3)-»«r+S)X.Ay)’}’. 
We  have,  moreover. 
and  thence 
^2 — ^3=  — 3AA, 
us  3 7Uj=  — SAB, 
ZiTj US2  — — 3 AC, 
A^+B^  + C^ 
% ^2  — ^3)\/  IH  — JU  = ^(<y  — 
(B-C)(C-A)(A-B) 
(/3  — 7)(a— 5) 
xO+a— /3— 7,  —hci-\-(3y,  M/3+7)— + 3/)"- 
And  taking  the  sum  of  the  analogous  expressions,  we  find 
a~^{[uS2 73-3)-y/lH ?«iJU  + (z<73 US^JV  + (j!Ji — OTjJU  } 
A2  + B2  + C2 
'(B-C)(C-A)(A-B) 
which  agrees  with  a former  result. 
(a— /3)(/3  - 7)(y — W 
