234 



Report S.A.A. Advancement of Science. 



ivhere /xi, //o- ••• i^ re the minors obtained by deleting the i^^, 2^^, ... 

 frame-lines of the qiiasi-pfaffian of O. 



For the sake of shortness in writing let the given determinants 

 be of the fifth order, viz. : — 



ai a2 CZ3 a^ a^ 



bi bo bs bi 65 



C\ C2 ^3 Q ^5 



di ^2 ^^3 '^4 '4 



ei e.2 t's t'4 er. 



■ /32 /33 /34 /35 



— /32 • 73 74 75 



— fis —73 • ^4 ^5 



— fii —74—^4 • £5 



— A5 —75 —^5 — E5 



and let the pfafiian minors of the latter, 



1 73 74 75 



^4 ^5 



be denoted by fii, jx^, •••, /"5 ; then the hrst term of the aggre- 

 gate referred to in the enunciation is 



(i\ ih iH fii fto 

 bi ■ 73 74 75 



Cl —73 • ^4 ^5 



di —74 —^4 . £5 



t'l —75 —^5 —£5 



which being a " bordered " skew determinant is by reason of 

 Cayley's theorem resolvable into 



f5 



and therefore into 



(ai/iti — 61^0 + ti//3 — dim + ^1^5) Hi- 



The other terms are similarly shown to be equal to 



{—ao/ni + ho^i — Cii^z + d^jJ-i — C2fXr^-iX2, 



( CZs/ii — 63/X2 + <-3/i3 — i-hl^i + ^S^^b)■^^S> 

 { — aifii + 64^2 — Qi"3 + <i'4A'4 — t'4/X5)-/^4, 

 ( ^5/^1 — 65/^2 + ''5iU3 — '^5^4 + ^5A'5) • M5 = 



