127 
Asin 16, the functions 4; = xj — x, are solutions of Xf —0 and of the system. 
If a matrix be constructed as indicated in 14 and 16, from the coefficients of the 
first r—2 equations, it will be observed that the independent determinants Ds, 
s=— 1,0,1....n—r), will be linear and homogeneous in yi with coefficients 
composed of functions of ¢j. Ds will then be solutions of all equations except 
Yf—0, which requires the ratios of yi to appear. Hence, the invariants may be 
written 
Oe Ce Dee Doe. (Jia 2 ony b= 0), 13. 5 an — rr). 
18 q, xq; x?q, .... x'q, p, xp-}cyq | 
; 13 : 
Here the complete system is 
Kee fs 1 at eh 
7 fi xi ==" (ic—0)1 £23). Xx Si m= 
1 ; 1 ; 
Vis: fix df - Le Sade 
= a lesa i rere — 
The solutions of W,f—0, Xf —0 are 
Yj =Y1— Yi, (= X1 — Xj 
Yf expressed in terms of 1), @ becomes 
with solutions 
Uk = $x:65, Vij: (¢;)°, (kK=3...n). 
The functions ux are solutions of the system. We find on introducing ux and 
vi as new variables in Wf, the partial differential equations 
Bi po Sie ee ode 
ee ede 
= 0) (=e Tr 3), 
whose solutions may be expressed as determinants D, of the matrix. 
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