26 
MR. E. B. ELLIOTT ON THE INTERCHANGE OF THE 
which together are equivalent to (19) and (20) together. From (23) it follows that a 
homogeneous function of x^, . . . transforms into an isobaric function of y^, 
. . . , and that, i and w meaning degree and weight respectively, 
'lx - ' 
while from (24) follows the equivalent fact that an isobaric function transforms into a 
homogeneous one, and that 
%V^ 
From (19) follows the especially interesting fact that, if a function of Xi, x^ x^, . . . 
is isobaric in x^, Xq, x^, . . . upon considering the weight of x,. to be r — 1, so also is 
the transformed function of y^, y.^, yg, . . . isobaric in the same sense and of the 
same w^ei^ht in yo, yg, y^, . . . 
Again the substitution m = 1, w = 1 in (13) and (14) produces for us 
Jr + li + *3 + • ■. = - 4 + 3Y.-3. A + 4Y ® A + ,, 
and 
CZ cl/ . [ /-j\ cl ■TT /T\ ^ I "VT /D\ 
+ '^^3:7:: + • • • = “ 1 ^3^ .TF + ^3' .yy + a; + ■ • 
dx.. 
dx, 
dys 
dyi 
. ( 25 ) 
> ( 36 ) 
whe 
re 
- yA Yg^n = 2y,y„ - 2 y,y 3 + y/, = 2y,y, + 2y,y„ . 
These two transformations have been obtained by Professor Rogers (see note to 
Art. 1). The second tells us that what he calls primary invariants in Xj^, x^, . . . 
have for their transforms what he calls secondary invariants in y^, y^, yg, . . . 
We might now consider the results of putting m = 1 and w = 2, 3, ... in (13) and 
(14). By this means the transformation of lineo-linear operators of two, three, &c., 
steps is effected. For the general case m = 1, = 'll the results are 
X 
^ V 
+ % 
dXji 
+ 
dx 
71 + 
d 
+ • • • 
8 
{n + 1) J 
q(ii+ + + 
^(/l + 1) 
(n + 2) 
dy. 
+ • 
7i + 2 
. (27) 
X, 
dx„ 
+ 
dx 
3(^0 
n + 2 
dxji 
+ 
+ 3 
(28) 
Perhaps the most interesting fact to be deduced from (25) and (26) is the trans¬ 
formation of {— 1, 1 ; 1, 1], the second annihilator Lt of projective reciprocants. By 
subtraction of (25) from (26), or directly from (15) 
(-1, 1; 1, 1}.,= 1-2, 1; 2, OR, 
