144 
We shall denote the parameters for x(2), x(3) .... by t, s,..... , respectively, 
and have series expansion for Xi(x(2)) in the form 
dt? 
Xj(x@) = K(x 4 xatp x” 4 2” go met? 
= X.(x ) + X;(x) x/at + {Xj (x). x/2 + X (x). Ale 
jee 
+ {Xj'(~).x + BX (x).x’x” + Xx)” | + 
+ [ Xyr(x).x* + OXY (x).x/2x” + BX j(x).2? + 
ada aN F dt* 
A ey aa KGa) a |e. as 
1 Dee 
with like expansions for Xj(x(3)),... in parameters s,...... Substituting these 
series expansions for Xi in the above determinant and subtracting vertical 
columns in a proper manner, we have 
V Wak seine Vole )P 5. Wo(x’)® oe. Ve ge) 
By Age ees Dep)? +. Ky e+... .. KET epee 
CA CRIS Vos @ OM V6 .€ 4 EO Seas OO ULOCC CO SHED OVC HOTT ESERD ER SD rad dso ess 6 Oa @ 6 4 a Sa EnEiS 
Siw eid) tie > woe» oe «ves 6S SS Wise 6p bd new Oe Se we 8 ew ee we wl we 8 ee 5 ale a ein YS Dial eTne 
Or disregarding x’, x’, ... which are invariant, and retaining only the ele- 
ments of lowest degree in dt, ds, ...... , we have 
Since x is also invariant, ~” a= aes, which would be the above determinant with 
x 
the last column changed to Vy 40 xrt 2 Saket Geet 
