128 
Hence, the point-invariants are 
Xj = Xk 
X, — X2 
Ug = ee 1D} (k= Fs. 2 aa ne ee 
Gx, X20, Se ED oe) eee . 
ro 
The solutions of the complete system 
= df at 
Wyf= 2 x SO (k=0, ti Sey Ki — we 
Lo: 
r a on df 
Yr= 2 {x i TIGA +ar—A] b=, 
may be obtained in a manner similar to 18. The solutions Oi, yj of Xf—0, Wi 
= 0, introduced as new variables in Yf, give 
= df df 
Uf ele ee ° ae Ns i pee jy lpmreaestacain) (CS 
Y 2s { 94 Ti eae }. ay b= 
with solutions 
Wj 
P td 
Uk = 9k : $2, Vi = log 9i—- 7 
@ 
700 3 eto hei p= ees, 12) 
Introducing ux, vj as new variables in Wf, and reducing, we find 
} 4 ’ ) 
Te df =e (t= 2) ar 
/ = te Ss — — 
Wit = aye t 2H Uk ; Jace (t= et 3) 
‘ 
whose solutions are ux, and the determinants D; of a matrix constructed as in 18. 
The invariants are, therefore, 
ie D,, (hk 3. 51 ht oe 
X,— Xo. 
2%). *| q, Xq, X°q, ..---. x*—4q, yq, P, XP |, 
r>3 
For this group 
a i df 
W f= di xt — =0, t=0,1,...r—4), 
1 
n n e n 
Yf= diy: ey Xx f= el Xf = >i ee 
1 dyi 1 dxi 1 dxi 
The last two equations show that the ratios of the differences of the x’s, say 
X, —Xk 
uk —= ———’ (k=8, .... n), 
XX) Xz 
