38 
MR. E. B. ELLIOTT ON THE I^^TERCHANGE OF THE 
i.e., let us in any a;-operator symbolize d jdxpq by rjP in any y-operator dldijpo by 
and in any ^-operator d/dzpj by r)^. 
We may in this way write (71) or (74) 
m {fx, V, V ; m, n, n'}, = yf {% ^ + ^oi '>? + %o ^^7 + %’7^ + • • • 
+ ^ {^10 ^ + 2=01 y + Zoo + Zn 5? + z^oo + • ■ • ; 
V'» 
+ ^ {^lo ^ + Zqi y + Zjo + Zji fy + Soa y- + . • •}“ 
= /x^" y"' + i y"' (^''') + T7”' + ^ ^ (^’''), . . (75) 
where ^ means the expansion in terms of ^ and y given in (65), and where the 
symbolization denotes that the right-band member is to be expanded in terms of ^ 
and 7 ], and to have each product ^p in its expansion replaced by the corresponding 
djdzpq, in order to produce the operator in 2 dependent which is represented by the 
notation on the left. 
Thus in particular, assigning to different pairs in succession of the three parameters, 
fx, V, V , zero values, 
m [1, 0, 0 ; m, n, n]-_ — .(76) 
m {0, 1, 0 ; m, n, n}~ — y"'^^(^“) = y"'. . (77) 
m{0, 0, 1 ; m, n, w'}, = . . (78) 
while 
[/X, r, V m, n, n'], — /x (1, 0, 0 ; m, n, n). + r {0, 1, 0 ; m, n, n'}. 
+ v' {0, 0, 1 ; m, n, n]~. . (79) 
Precisely similar symbolical expressions to (75) . . . (78) are, of course, assigned to the 
corresponding operators in x and in y dependent. We have only cyclically to inter¬ 
change f, y, I, once and twice respectively, and to regard the expressions on the right 
thus obtained as short ways of writing their expansions by aid of (63) and (64) in 
terms of y, { and ^ respectively. 
17. In the present article the expression of each operative symbol djdx.s, on a 
function of the derivatives of x with regard to y and 2 , in terms of the operative 
symbols djdzp^ on the equivalent function of the derivatives of 2 with regard to 
X and y, is investigated. 
