462 
ME.  A.  CATLET  ON  THE  TANOENTIAL  OF  A CUBIC. 
so  that 
g,  Xy.z), 
s=(/,  c Xy,zf, 
C:^{k,  U g \y,z)\ 
D=(5,  /,  i,  c\y,  z)\ 
{h,  b,  i,f,  I,  k X^,  y,  zf={h  , P,  Q ^x,  If, 
g^  ^ y^  ^f=U  ^ s x-^^ 
(«,  h,c,f,  g,  h,  i,j,  k,  =(« , B,  C,  DX^,  If- 
C^+3B  =(C,  B X^’>  1)^ 
and  then  for  greater  convenience  writing  (A,  2P,  (XX.x,  1)®,  &c.  for  (A,  P,  QX^t,  If,  &c., 
and  omitting  the  {x,  1)^,  &:c.  and  the  arrow-heads,  or  representing  the  fnnctions  simply 
by  (A,  2P,  Q),  &c.,  we  have 
.r^|=  A(X  2R,  S f 
-3/(X  2R,  S f.(A  , 2P,  Q) 
+ 32-(X  2R,  S ).(A,  2P,  Qf 
— c , (A  , 2P,  Qf 
- («,  3B,  3C,  D).(C,  B ), 
which  can  be  developed  in  terms  of  the  quantities  which  enter  into  it.  The  conditions, 
in  order  that  the  coefficients  of  x'^  may  vanish,  are  thus  seen  to  be 
DB = AS*-  3/S^Q+ 3iSQ^  - cQ*, 
DC  - 3CB = A(6RS^)  - 3/(2ST -f  4RSQ}  + 3?(2RQ^ + 4SPQ)  - c6PQS 
and  from  these  we  obtain 
and  substituting  these  values,  the  right-hand  side  of  the  equation  dindes  by  .r*,  and 
throwing  out  this  factor  we  have  the  value  of  | ; and  the  values  of  n,  ^ may  be  thence 
deduced  by  a mere  interchange  of  letters.  The  value  for  | is 
