86 
Proceedings of the Royal Society of Edinburgh. [Sess. 
thus must disappear is 315 ; and, this not being a multiple of 90, it cannot 
be that their aggregate is representable as a sum of vanishing determinants 
— at any rate, of determinants that vanish because of having two rows 
alike.* 
(8) The occurrence of Pfaffians with polynomial elements of special form 
naturally turns the mind to polynomial elements that are unspecialised, 
and to the question of the existence of an analogue to Albeggiani’s or other 
like theorem in determinants. Such a theorem, when freed of details, is 
the following : — Any 2n-line Pfajfian with k-termed elements is equal to 
the sum of k n "Pfaffians with monomial elements. The simplest case is 
in full 
I flq "I - oq 
= a. 
a 0 (Xq 
h 
Co 
P~2 
a 2 + a 2 
a 3 + a 3 
&2 + P 2 
b 3 + /? 3 
C 3 + 73 
% 
+ 
cq a 2 , 
P3 
73 
+ 
A 
a 3 
Pz 
73 
If we view the triangular array of such Pfaffians as composed of two 
parts, and use i to stand for “ Italic ” and g for “ Greek,” this expansion 
may be greatly abbreviated, — thus 
ii + ig + gi + gg . 
Taking a further step towards ultra-symbolism, we may use for this 
(i + g)(i+g), 
or even, with a proviso, ( i+g ) 2 . When the elements are trinomial the 
result is 
J Ciq + cq + Aj 0>2 + a 2 + Ag Cfcg + a 3 + Ag 
A + P2 + A ^3 + Pz + ®3 
C Z + 73 + ^3 
= ii + ig + ic + gi + gg + gc + ci + eg + cc 
= (i + g + c)(i+g + c) = ( i + g + c) 2 , 
where, in similar fashion, c is used for “ capital.” 
* This is at variance with a statement of Pascal’s when dealing with Brioschi’s theorem 
above referred to. In his paper on “ Un teorema sni determinant di ordine pari,” he says, 
“ Gli altri termini dello sviluppo del prodotto (7) sono similmente termini dello sviluppo 
di determinant come D dove pero alcune coppie di linee sono ripetute e che quindi 
sono zero” (see Rendic Accad Napoli , xxv, 1919). The case where m is 2 in § 3 
provides the simplest test : for then the number of terms on the left is twelve, and the 
number on the right is 6, and the 6 on the left whose aggregate must vanish are 
| I • | «3&4 I , “ | | • | «2&4 I , I <hh I • I ®2&S | , 
| Old 2 I . I c 3 d 4 | , - I c x d 3 | . | C 2 d 4 | , | c x d 4 j . J c 2 c ? 3 | , 
which, instead of forming a vanishing determinant, form two vanishing Pfaffians. 
