145 
13. X14 Xa, secrete Doel ¢ yq | leaves invariant x and the ratio 
Po : Py where dy, dy are determinants defined in 12. 
Since x also remains invariant, we may write our differential invariants 
14. The special linear group 
|_P, 4 xq, XP—yq, YP | 
has the point-invariant 
x y 1 
=x y" 1 
x?) y°” 1 
Expressing x), y x) y”) in series expansion in terms of t, s, 
x y 1 
7 ee die yy Py dts Sa acy E 
2 2 
rex! dae yty’ds+y” ce Sea tek 
dt ds? dt ds* dt ds* | dt? ds? dt ds® 
= Eps — EN fe pees 
Eo NL ane OAT, aman a ine eas 
2 4 
oa dt? ds 
48 
De Ye XN — Yo) aia Xt PE”, 
Pr ng ees (Ela eee yom =| overs Ce ty (a)? x7, 
i; — x/// y” — x y//’ =y, [ (x’)2 x’ — 3x’ (x’’)?] =. (x’)8 x 
I, = x’ yY— x’ y’ =y, (x’)® + 10y, (x’)* x” + 15 (x’ x”)? + 10y, (x)? x” + 
=P ly; xxx (e- by es" 
I, =x” yiv— xiv” = y, (x’)* x” + by, (x x”)? + yo [3 (x)? + 4x! x” x — 
x5 (4) xiv i 
5 
10—ScIENCE. 
