ON PFAFF-INVAMANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxvi. (1893), 
pp. 195—205.] 
1. The functions which I propose to call Pfaff-invariants present themselves and 
play a leading part in the memoir, Clebsch, “ Ueber das Pfaffsche Problem ” (Zweite 
Abhandlung), Crelle, t. lxi. (1863), pp. 146—179 : but it is interesting to consider them 
for their own sake as invariants, and in the notation which I have elsewhere used for 
the functions called Pfaffians. The great simplification effected by this notation is, 
I think, at once shown by the remark that Clebsch’s expression R, which he defines by 
the periphrasis “ Sei ferner R der rationale Ausdruck dessen Quadrat der Determinant 
der a ik gleich ist” (l. c. p. 149), is nothing else than the Pfaffian 1234... 2n — 1.2n, 
dR 
and that its differential coefficients Ri k = »— are the Pfaffians obtained from the fore- 
da^ 
going by the mere omission of any two symbolic numbers i, k. 
2. I call to mind that the symbols 12, 13, &c., made use of are throughout such 
that 12 = — 21, &c. ; and that the definition of the successive Pfaffians 12, 1234, &c., 
is as follows : 
12 = 12, 
1234 = 12.34 + 13.42 + 14.23, 
123456 = 12.3456 + 13.4562 + 14.5623 + 15.6234 + 16.2345, 
in which last expression 3456 denotes the Pfaffian 34.56 + 35.64 + 36.45, and similarly 
4562, &c. ; and so on for any even number of symbols. Of course, instead of the 
symbolic numbers 1, 2, 3, &c., we may have any other numbers (0 is frequently used 
in the sequel as a symbolic number), or we may have letters or other symbols.