1208 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
*^2^6 4 ’ 3 ^ 6 ^) {(^2 ^(^sLg) U-^ ^ — 0 . . . 
where all the functions (p contain differential invariants only. Such a line may be 
called homographically persistent with regard to the standard curve. 
Indicating a second homographically persistent line by a similar notation, but using 
dashed functions, we may find the coordinates of the point of intersection in the form 
where 
77 = u^Kju^k. + C + Bctg 
^ = u^ju^A. + C + Bag 
A = — — (pzh) 
B = 7^5 {{(ps<Pi — (p3(pi) — {(p-zh' — 
c = {(pi<p.2 — (pii>2) + 2 {(p^(pi — fp3<Pi) Le — — ^2^z) W.. 
>3 
or, more shortly. 
A = — 
B = u.^ {ix — XLg) 
C — V fi- 2p-Lg — XLg*' 
(56). 
We may call such a point a point of homographic persistence with regard to the 
standard curve. 
Treating {XijjLw) as determining the position of a point, and {(pi ■ (p-z ' ^ 3 ) 
position of a line, the condition that (X : p. : v) may lie on {(p ^: (pn : (p^) is 
X(/)i + fX(p .2 + V(p.^ = 0. 
Hence these form correlative systems of point- and line-coordinates. 
If two lines {(p^ : (p ^: ^g), {(p\ : (p'^ ‘ <p'z) meet in (X : p, : v) 
X /X V 
P' 2 ^ 
Pz 
\ P3, 
Pi 
\ Pv 
p2 
P'2> 
p's 
1 J / 
i 935 
p'l 
1 P'v 
P'2 
and if two points (X : p, : v), (X' : /a' : v') lie on {(p^ : (p^: (p^) 
Pi Pz 
1 V, X 
X, 
1 f ' 
1/^3 V 
v', X' 
1 ’ 
V, 
A : 
