32 
MR. E. B. ELLIOTT ON THE INTERCHANGE OE THE 
or, replacing vm by v, 
m |/x, - ; m, — wX = ~ [I — m) \v, — ; 1 ~ m, — 11 
[_ in \ X LI 'Wl \ y 
+ (P- + 2 / 1 ”'" {O 5 1 j I 5 ~ 1 ly 
In particular, 
m 
^ " ; 1 ~ w, m 
1 “ m 
(45) 
(4G) 
12. The value zero of m is somewhat special in these cases of n = ~ m, just as in 
the more general cases already discussed. So too, of course, is the conjugate value 
m = 1. 
For (43) and (44) to hold for these special values of m we must have. 
0 ( 1 , 0 ; 0 , 0 ]^ = - 
{1, 0; 1, - 1}, = - 
{ 0 , 1 ; 0 , 0 ). = - 
{ 0 , 1 ; 1 , - 1 ],= - 
{ 0 , 1 ; 1 , _ 1 ]^+ { 0 , 1 ; 1 , - l}y, 
{0, 1; 0, 0], 1; 1, - 1],, 
{1, 0; 1, - 1],+ (0, 1; 1, - l\y, 
0 {1, 0; 0, 0], (0. 1; F “ Ik 
(47) 
Of these four equalities the first is a mere identity of two zero operators. In fact 
to 0 {1,0; 0, 0] no other meaning than zero could be attached consistently with the 
general definition. Thus the form of {1, 0 ; 0, 0] is left indeterminate. 
The fourth of (47) becomes 
^.e., 
{0,1; 1, = [0, 1; ], -1]„ 
}=?/r*{2y.|; + 3ya4 
^ d , „ d , , d . 
2(^0 -fi 3 X 0 , “b 4iX^ -z -j- . 
^ dx^ ^ dx^ ^ dx.^ 
(48) 
d . ^ d 
of which the left-hand member is merely x-^~^ —, and the right y~^ —, the symbols of 
<7 
differentiation being total. 
The remaining equalities, the second and third of (47), now become the same but 
for an interchange of x and y. Consequently there will be complete consistency if 
we define the at present undetermined operator {0, 1 ; 0, 0} as that which obeys the 
equation of transformation 
{ 0 , 1 ; 0 , 0 }, = - { 1 , 0 ; 1 , - 1 )^- 1 - { 0 , 1 ; 1 , - 1 ], 
= { - 1, 1 ; 1,-1],, 
(49) 
