664 MB. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
$4. 
The resultant equation which, when combined with that of the original curve, will 
determine the sextactic points, was exhibited in § 2 under six different forms, there 
designated by 
1=0, iH=0, #=0, £'=0, iW=0, #=0. 
Now since % and %!, i'H and XW, iX and -ffij respectively differ only by the numerical 
factor (n— 2) 3 , we shall, in seeking to discover the extraneous factors, employ either 
S., . . or ■%!, . . as most convenient for the purpose. And in the first place it will be 
shown that H is a factor of all these expressions. Putting H=0, %! becomes 
o x uX b x uY b x uZ u — 0 ; 
b y uY b y uZ • v ....... (43) 
b z uX b z uY b z uZ w 
A uX AuY AuZ gt 2 H 
also 
AwX=j)X+mAX+2HB # X I 
AmY=jpY+wAY+2HB,Y (44) 
AuZ =pZ -\-uAZ +2Hd x Z ; | 
so that the above equation, written in full, is 
u Y X +wb^X UjY -f-wd x Y w,Z -\-ub x Z u 
w'X +wb ? X w'Y +mc^Y w'Z -f- ub y Z v 
v’X+ub z X v'Y +ub z Y . v'Z+nb z Z iv 
p X +uAX+2Hb x X p Y +mAY+ 2HB x Y p Z +uAZ + 2llb x Z ar 2 H. 
Although this expression contains terms explicitly multiplied by H, which might on 
the present supposition be omitted, it will still perhaps be worth while to develope it 
completely. Expanding in the usual way, it becomes 
u*X u x <3,Y bJZ u +w 2 Y u l bJZ b x X u +u 2 Z u, b x X b x Y u +u 3 b x X b x Y b x Z u 
V) ' b,Y byZ V W' byZ ^ ,X V ^ b,X V d,X b y Y V 
v' b z Y b z Z w v' b z Z b z X w v' b z X b z Y w b z X b z Y b z Z w 
p AY AZ *t 2 H p AZ AX st 2 H p AX AY sr 2 Ii AX AY AZ 
+H u^X-\-ub^X MjY+wB^Y u{L-\-iib^L u 
w'X+wc^X w’Y+ubyY w'Z-\-ub y Z v 
P X + wb z X v'Y-\-ub z Y v'Z-\-ub z Z w 
2 b,X 2b x Y ' 2b x Z sr 2 . 
