ME. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OE A PLANE CIIEYE. 661 
and noticing that 
X-dj-Ou - } - !! B y Qu-{-/3 z Qu — Jac. (U, H, Qu), 
. . . (35) 
and that 
AX= vd z Q v —wd s O 0 1 
AY -wd x £l v - dd., O rJ 1 
AZ = wd^Qu— wb^Ou, | 
. . . (36) 
then we have 
YAZ -Z AY —u Jac. (U, H, tjAZ -z AY=(5w-12)Qm 1 
Z AX— XAZ s=v Jac. (U, H, Q n ) zAX-xAZ =(5n-l2)Qv i. . (37) 
XAY— YAX=w Jac. (U, H, Q D ) ®AY— yAX=(5»-12)Qw.' I 
Again, if 33', C', 4F, 0', be the same quantities with respect to H that 
3, 33, C, jf, 0, are with respect to U, i. e. if Q!=q l r l — p' 2 , . . jf l =q'r'—p l p', . . , 
and if 
©=($', 33', C', f, 0', H')( w > wf ] _ ^ 
^=(& 33, C, f, 0, H)(ih q , r) 2 , 
then 
Uy 
c^Z=Jac. ( u , Y r , Z,)=w, 
if p — u x r + wp x — uqf 
w' 
^Z w' 
dp—w'r-\-wif —up 1 
if 
d z Y 
d s Z «/ 
Wy p—v' r-\-wq' —ur x 
=Hp 2 
— Hp 2 — Qpu + (%{p j + $|r' + 0#' )pu 
— (!?*' + 33^1 +Jfp')pw 
+ (11^1+33/ + Jfy' )pv 
+(<®Pi+$ r r' +€q')wp 
+(0^i+ + CV )ww 
-(02' +4^ +Cr» )pt* 
+( 91 'mi +H f w'+ 0 V)w 2 
+ (H'^i+33V+ f'v')uv 
+(Hih + 33/ +jf2 / )jw 
+( 0 Pi + 4 ^ 4 -C^' 
+(&'%, +^V + 0 V)m 2 
+(H'wi +35 V -j-ffi'v'juv 
+ ( 0 'w i + 4 f V + C )mt? 
Similarly, 
Jac. (w, Z, X)=w t ■w/p+wr' — w'r—v 1 q-\-vq ’ — wr' 
■y' w' ^ — dp + up' —vq' dr — Wjy-f- Wj — wp 1 
4 Y 
MDCCCLXV. 
