254 
ON THE SEXTACTIC POINTS OF A PLANE CUPcVE. 
[341 
Moreover 
and thence 
TU= (a, ...'fyvdy- gd z , ...)• TJ, 
= a ( bv- + eg- — 2//jlv ) 
+ b ( cA 2 + civ 2 — 2gv\ ) 
-I - c ( ci/jL- 4- b~X- — 2/iA/i) 
4- 2f (— fX 1 + r/A/i + h\v — cifjLv) 
+ 2g ( fXg — gg- + hgv — bv A ) 
+ 2It ( fvA + gvg — hv 2 — cA//,); 
F3U = (a, .••'§vd y — gd z , ...)-dU, 
= a (v-db + g 2 dc — 2gvdf) 
that is 
Hence the equation 
becomes 
that is 
4- &c. 
= A 2 (bdc + cdb — 2fdf) 
4- &c. 
= (321, 323, 3®, 3& 3®, 3£$A, ya, *,) 2 , 
T3H = 34>. 
TC3H- V 2 3H=HT3H 
03H- v-dU = Hd<s>, 
V 2 3 Z7 = $3 H — Hd<&. 
69. Proof of equation 3 X 3 2 2 H = — Hd c £>). 
We have 
^ = ( m -T) 2 ^ + y0? ' + ^ 
“ (7^1 ) 2 ^ + y d y + zdz) V 
^ (m - l) 2 ’ 
and thence multiplying by d x , =3, and with the result operating upon 
3i3 2 2 u= 
(m 
-■<p-'3 [/■ - ^ M>3 V H + — 
(m-1) 2 ° (m-1) 2 ^ ° + (w-l) 2 
U, we find 
3 V 2 H. 
But dU=0, and thence also V (3U) = 0, that is (V .3) U + V3U = 0 ; moreover V . 3 = 0, 
and therefore (V . 3) U— 0, whence also V 3 IT =0. Therefore 
a i 3 2 2 H= / —3V 2 H; 
(m — lp 
or substituting for 3V 2 C/ r its value =<$dH — Hd®, we have the required expression for 
3i3 2 2 Z7.