106 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
Now writing for greater simplicity XY instead of X.Y, and so in similar cases, we 
have, as regards Y*A, 
XY-YX=^-2.9. 
Hence 
and consequently 
Similarly 
and therefore 
(XY-YX)A=^A, 
XYA=YXA+M=i^A. 
{XY-YX)\A=]:^2YA, 
XY^A = YX Y A +^'^2 Y A 
=-[iYA-{-(ju — 2YA=2(|M/— 1)YA, 
And again, 
and therefore 
or generally 
(XY-YX)Y^A=|t4-4Y^A, 
X Y* A = YX Y^ A + ^ - 4 Y^ A 
= 2 (^— 1Y'A+|«.— 4Y"A=3(|!a— 2)Y^A, 
XTA=s(iJ.-s+l)TA. 
Hence putting A==/a-+l, jt/-+2, &c., we have 
XY'^^'A=0 
XY"-^^A=-(p+2)l.Y'^^^A 
XY"^*A=-(^+3)2.Y"''^A 
&c., 
equations which show that 
Y'^^^A = 0; 
for unless this be so, i. e. if Y''’^'A=|=0, then from the second equation XY'"'^^A=|=0, 
and therefore Y'^^^AH^O, from the third equation XY^‘^^=l=0, and therefore Y'‘^^A=1=0, 
and so on ad irifinitum, i.e. we must have Y'^’^'A=0. 
31, The suppositions which have been made as to the function A, give therefore the 
equations 
XA=0 
XYA=^A 
XY^A=2(y.-l)YA 
XY'^A=^.Y^-'A 
Y'^-^’A=0. 
B=YA, C=iYB, ..A' = iYB’, 
And if we now assume 
