THE  EEV.  G.  SALMON  ON  CEEYES  OF  THE  THIED  OEDEE. 
539 
and  those  of  the  second  tangential 
— y){/3(y— a)"— 7(a--/3/}  ; Y=^(y~a){y(a— /3)"  — a(/3  — y)=*} 
Z=z(cc-(5){oc(^-ry-(3(y-ay}. 
In  calculating  the  expressions  for  X and  (m,  I have  found  the  following  theorems  useful. 
Let 
(/3— y)"+  (y— «)'+  (cc—(5y=2A, 
a (/3~y)"+^  (y“a)"+y  (a— /3)"=B, 
cc%(3~ry+(i%r~ay+f(c~(3y=2C, 
(a  - /3X^  - y)(y — os) = Q. 
Then 
(i3-y)^+  ir-«y+  (^-(5y=2A\ 
a (/3— yj^+ZS  (y— a)'+y  (a— /3y=:AB, 
a\(5-ry+(i%y-ay-\-f(a~(5y=2AC-Q^; 
and  if 
p=a-f/3+y,  2'=a^+^y+ya, 
we  shall  have 
(/3— y)®+  (y— (a— /3)®=2A® 
« (/3— y)®+/3(y— a)®+y  (a-/3)®=A"B  +^Q^ 
u%^-7y-\-(^%7-czy-\-f(oc-[3y=2A^C-{f-4:q)Q\ 
(i3-y)^+  {7-o^y+  (cc-f3y=2A*  ^-8AQ^ 
« (i3-y)^+/3  (y-«)*+y  (cc-(3y=A^B  +(2pA+B)Q^ 
cc%f3-7y-^f3%7-cy+7%cc-f3y=2A^C-y(2sA-y2C-A^)Q^; 
while  again, 
(^_y)3+  (y_«)3-p  («_^)3=3Q, 
a (f3-7y+(3  (y-ocy+7  (a—f3y=j)Q, 
u^(f3-7y-y(3%7-^r+f(^-f^y=sQ, 
((3-7r+  (y-«)3+  («-/3)3=:5AQ, 
“ (f^—yy+f^  (y— a)'+y  (a— /3)®=(pA+B)Q, 
uXf3-7y+f3^(7-^y+f(^-(^y=(s^-y2C)Q. 
By  the  help  of  these  equations  I obtain 
— j«/ = (B — 2Alxyz)Q 
\ =(B-  2A%^)A(AC-  Q^). 
We  may  of  course  suppress  the  common  factor  B — 2Alxyz,  which,  by  the  help  of  the 
equation  of  the  curve  'p-\-%lxyz=^,  is  seen  to  be  equivalent  to  —^{l-\-M^)xhfz^-,  and 
the  coordinates  of  the  required  point  are 
