1919-20.] Note on Pfaffians with Polynomial Elements. 87 
For the Pfaffian of the next higher order we have 
j ttj + Ctj ^3 a 3 ^4 a 4 ^5 "t" 
^2 + ^2 h + @3 ^4 + ^4 h + Pd 
c 3 + y 3 c 4 + y 4 c 5 + y 5 
^4 + ^4 ^5 + 
e 5 + e 5 
= iiii + iigg +■ igig + iggi + gggg + ggii + gigi + giig 
= (ti + 99)* + (ig + g$f. 
i 
9 
i 
9 
i 
9 
i 
9 
9 
i 
9 
i 
and when the elements are trinomial 
(u + ^ + cc) 2 + + gc + m‘)(w + gi + eg) + (ic + gi + cg)(ig + ge + ci) 
i g c 
i g c 
i g e 
i g c 
9 c i 
c i g 
c i g 
9 e i 
(9) It has to be observed that in the foregoing theorem the degenera- 
tion from elements of k terms to elements of k — 1 terms does not take 
place with the readiness found in the case of the analogous theorem 
in determinants. Thus the twenty-seven Pfaffians in the example just 
given do not, on substituting zeros for the capital letters, at once reduce 
to the eight of the preceding example — a fact which is manifest on noting 
that two of the eight, igig and gigi, do not appear among the twenty-seven 
at all. What actually takes place is the immediate vanishing of fifteen of 
the twenty-seven, and the partial vanishing of six, the latter group being 
igi . -J- ig.g + i.ig + gig . + gi.i + g. gi , 
which condenses * into 
igig + gigi 
as it ought. 
(10) The like is true in regard to degeneracy from Pfaffians of one order 
to those of the next lower order. For example, if in the last equality of 
§ 8 we equate cofactors of e b in the hope of obtaining the second equality 
of the same paragraph, we are disappointed ; an additional transformation 
dependent on the manifest equality 
A 4 A 2 A 3 
+ j A 4 A 2 A 3 
— j A 4 A 2 Ag 
+ | Aj A 2 Ag 
B 2 1*3 
^2 ^3 
B 2 Bg 
^2 ^3 
c 3 
C 3 
C 3 
C 3 
is necessary. 
* Trans. S. African Phil. Soc xv, pp. 35-41, §§ 6, 7. 
