656 ME. W. SPOTTSWOODE ON THE SEXTACTIC POINTS OE A PLANE CURVE, 
so that (7) finally takes the forms 
(a <B, C, Jf, 6, a.) 5 V- (l+^^)«H=0 .... (14) 
or, in the case where V is a conic, and consequently m— 2, 
gffl+3BJ+Co+2(4r/+%+»7 i )-i^ildH=0; ..... (15) 
2 [m i) 
and in general making sr=l+ n _ f , (14) takes the form indicated above, viz. 
AY— w0H=O, v 
° r A Z [ ( 16 ) 
u v w otH J 
§ 2. Elimination of the Constants of the Conic of Five-pointic Contact. 
Before proceeding to the application of the formulae (16) to the investigation of the 
sextactic points, it will be convenient to premise that if s, t be any two homogeneous 
functions of x, y, z, the nature of the operation A is such that 
Astf=sA#+tfAs+2(&, 35, C, tf, (B, ||)(b,s, b/, d z s)(dj, df, d z t), . . (17) 
and also that 
AV=3H, A u=_p, Av=y, A w—r (18) 
This being premised, our first object is to establish an equivalent for □ 3 V=0, divested 
of the extraneous quantities a, (3 , y. Now, since 
X v'd z Y-w'd y Y)=xDY, 
Z(w\Y-ud M V)=yU Y, 
Xu'dff- vb,V)=snV, 
and DS=0, it follows that 
i □ ( vb z Y - wd y V)=*. □ V +ar □ 2 Y, 
S □ (wb x Y — uby)=(A □ V + y □ 2 Y, 
&D(wb,V_ »b,V)= *DV+zp 2 V; 
and consequently not only do vb z Y — wb y V, vfd x v — ub z Y, ud y Y — vb a Y vanish with mV, 
but, when this is the case, □ (yb s V— wb^Y), .. vanish with D 2 V. The same will 
obviously be the case if the operation □ be continued ; so that, in general terms, we 
may, by operating upon vb^—w'b^Y, . . with the symbol □, 0, 1 , 2, . . times, form a 
system of equations equivalent to that formed by operating on V with the same symbol 
1 , 2, 3 , . . times. And if we represent any of the three quantities vd z Y — vfb y Y, . . by W, 
the equations W=0, DW=0, •□ 2 W=0 will be equivalent to the system 
b«w a y w_a,w aw 
u v w ra-jH 5 
( 19 ) 
