MR. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CURVE. 657 
analogous to (16). More generally, if 
Ay=u A — xffHd,, 
A 2 =v A— arHdj,, 
A 3 =wA — zzrHc)*, 
and if A' stands for any of the three symbols A,, A 2 , A 3 , then the equations V=0, 
□ V = 0 are equivalent to 
-B jr y=-B w y=-B s y; 
U * V y w z 
the equations CPV^O, 0^=0 are equivalent to 
-^A'V=-d A'y=-^A'y. 
U x v y W z 
Similarly, if A' 7 stands for any one of the symbols A,, A 2 , A 3 , either the same as A' or 
not, then D 4 y=0, [H 5 V=0 are equivalent to 
- d. A" A'V =- A" A'V = - A" A'V, 
U x V y w z 
and so on indefinitely, for □ 2i V = 0, □ 2i+i y=0. If the series should terminate with 
□ 2i V=0, e. g. D 6 y=0, then the last equivalent would be A"'A"A'V=0 , where A!" 
stands, like A", for any one of the symbols A M A 2 , A 3 indifferently. The form W, 
however, presents peculiar advantages for the application of the operations A, as will 
be more fully seen in the sequel. And it follows from what has been said above that, 
if W retain the same signification as before, we may replace the equations W = 0, 
□ W=0 (and consequently the equations Qy=0, D 2 y=0) by 
-d,W=-^W=-B,W, 
u x v y w z ’ 
and in the same way the equations Q 2 W = 0, □ 3 W = 0 (and consequently □ 3 y = 0, 
□ 4 V=0) by 
- A'^W = - A'S,W=- A'B.W, 
U x V y w z ’ 
and so on. I do not, however, propose on the present occasion to pursue the general 
theory further. 
Returning to the problem of the sextactic points, and forming the equations in W 
(19), we have 
h x (v B,y- w^y)=^> wB,y)=^B> ^v- wB,v)=^ s a(v z z v- w^v) ] 
^B>^y-MB,y)=^ y (wB,v-MB z v)=:^,(wB,v-w5,y)=^| I A(wb,v-wS,y) . 
ld z (u d y V-v *.V)=\b,(u t, J f V)=^« d y V—v d z V) A(« B,V). 
( 20 ) 
