129 
shall appear in the final solutions. The n —r--3 independent determinants 
Ds, (§ = 0,1.... n—r-+ 2), of the matrix 
Vi Vo Wag 44 3s Vee cae tel TH 
1 it Poca! Paras saa a ate ahs 1 
Xy Xo See kas ya Lay REO. ee 
ae Novae Mae Set onis = chs 2, 
are solutions fof the first r—3 equations Wif—0. These determinants are, at 
the same time, homogeneous in yi and in xi— xx; their ratios will, therefore, 
satisfy the requirements of ux and Yf—0. Hence, we may write our 2n—r 
invariants as Ux— (x, — Xx) :(x,; — x.) and Rt: = Dr: Do, (k= 3...n, t=1... 
n—r- 2). 
- 
91. | % *4 x°q, .... x*—4q, p; 2xp-+- (r—4) yq, x*pq- (r— 4) xyq . 
r>4 | 
From this group we obtain the differential equations 
n * 
Ww if : 
eats eb 0) (V0) Le ==) Kfsi# =o, 
t dyi J : 
1 axi 
ae ere BS ge pe 
XxX rer ears sig Chae ee mo” 
Xf = Si § x32 a = (r— 4) xi yi | ==\)- 
rt eh bg eh dyi 
The solutions of Wof = 0, Xf—0 are ¥j—y, — yi, 4; =x, — Xi, respectively. 
X,f when expressed in these new variables becomes 
n 
It ——= Ss; a2 af — = aps a —— 
AX,i= 2: {4 doi + (r 4) $j Yj ade 0, 
whose solutions may be selected in the forms 
1 1 ; 
ae es —— w=, (k=8... An os 
— df 
x i232 Sem ge +O NG, =O 
du ‘dvi 
has solutions 
oli: ug, %& = Ve: un @—, 6, =v,:u, %—-%, (k= 3.... n, 1=4....n), 
9—SCIENCE. 
