30 
MR. E. B. ELLIOTT ON THE INTERCHANGE OF THE 
of pure reciprocants is, when affected with a simple multiplier, self reciprocal. We 
may write (34) 
d d 
= - yi 0, 1 ; 1, 0},, - V (y, x). 
• ■ • L, “ ~ ~ • 
So too, correlatively, 
yi~ 7 — ^i~r 
^ dy-^ ^ dx-^ 
{1, 1 ; 1, 0}^ — .Tj W {x, y) 
But 
(1, 1 ; 1, 0].,= - {1, 1 ; 1, Oj^by (18) or (20) 
. •. rrj W (a;, y) = - yi W (y, a;);.(38) 
so that, to use a familiar notation, ^“W is a self reciprocal operator of negative 
character. 
11 It remains to consider operators {p, v, m, n] in cases when mn <1. In 
such cases the formulae of Arts. 5 and 6 have to be replaced by others. The essential 
difference between them and the cases already considered lies in the fact that the 
lower limit of in (3) and (4), and, therefore, in what replaces (7), is now — 71+1 
in.stead of m, i.e., is greater than m, so that the coefficients in [fx, v ; m, n},j are no 
longer multiples of the complete set of coefficients in the expansion of (y^^ + y^P 
+ + • • •) but of those coefficients with the exception of one or more at the 
beginning. 
In the present article attention is confined to the case of m + = 0, i.e., n= — m. 
Proceeding to write down the symbolical form of {y-, ; 'in, — '>n}y as in Art. (5) we 
see that the whole expansion from which we there started is present except the first 
term, 
-(y + rm) + 
Thus the symbolical form of 
{1,0; 'ill, —'m}y is ^ (y"'— yf" .(39) 
and that of 
{0, I-, 'ni, — m}y is — yh'. . (40) 
the right-hand members standing for their expansions in powers of f 
