50 
MR, E. B. ELLIOTT ON THE INTERCHANGE OF THE 
it follows at once that 
~ {x, yz) = — - ^o{y, zx) 
'-01 
Vv) 
= -2,JO, 1,0; 2, -1, 0].-l-2io{0, 0, 1; 2,0, -ij.,. (119) 
which, and its correlatives, obtained by one and two cyclical transformations effect 
the transformation of and 
Again, we may write (114a) 
dx 
d . d 
= - (y, zx) - y^] , j 
01 
= — VJz, xy) — Zqi (z, xy) — Zio{l, 0; 0,b- + ^ 
10 
dz 
10 
and (114) in like manner 
-- = — VJy, zx) — yioii 2 (:i/> ~2/oilk 0, 1 ; k o, o j + yoi 
Ctj'YQ ^ 01 
= — Hi (z, xy) — z 
10 
dZr, 
From these it follows at once that 
d 
^’lo 7,7" + ^’oi J 
ttiLj 
01 
l/io 7^ + !/o] 7^ “ -^2(1/7 - {1, 0, 1; 1, 0, 0}, 
dy^o dyoi ^oi 
= z7o/- + ^oi7^-r"Vi(^^^3/)- {k kO; 1, 0,0b 
“Zjo «-01 ~10 
By a cyclical interchange of the variables we have, also. 
X 
iu 
'■uiJ- bLG 2/“) - ! *, 1 > 0; 1 , 0 , = + 
d_ 
d//in 
A 
dUoi 
— "lu 7 , + ^01 7 
“•‘lO 
- ik 0 , 1 ; 1, 0 , Ob; 
01 ^01 
and by a second cyclical interchange a, third set of such equalities is obtained, hiom 
the two sets that have been wuitten out, upon subtraction we obtain 
— Vi {x, ijz) + { 1 , 1 , 0 ; 1 , 0 , O 5 
■'^-10 
Un 
2 ^) - ik 0 , 1 ; k 0 , 0 ], 
VJz, xy) + — Vo (z, xy) + fO, — 1, 1; I, 0, 0]; 
'•10 
