Determinants and Pfaffians. 



235 



and thus the sum of them all is 



— ^2 



— hi 

 "5, 



as was to be proved. 



3. // P be any n-liiie detenninant, Q a zero-axial skew determinant 

 of the same order, and Ar a determinant formed by replacing the r^^ 

 column of Q by the r^^ column of P, then when n /5 even 



r=l 

 S being an aggregate which vanishes when P is axisymmetiic. 



If for the purpose of illustrative proof we take 



P = I rti 62 fs di e^ /e 



and Q = I «2 "3 "4 "5 "6 



/33 /34 /35 /Se 



74 75 76 



^5 ^6 



the first term of SAr- is 



«6 



«i «2 "3 04 05 ae 



61 • /33 /34 A5 /36 



<:i —fiz ■ 74 75 76 



di — /34 — 74 . O5 ^6 



£\ —Po —75 —4 • f6 



/l — /^6 —76 —^6 — f6 



which by Cayley's theorem previously referred to is resolvable into 



and therefore 



I "2 "3 "4 as "6 



/33 /34 /35 /36 

 74 7o 76 



f6 



^'1 



Ih fh 1% l\ 



74 Jb 76 

 ^6 



+ ^1#12 — Cl#13 + ih-ffu — f]#i, + ./l#16J 



R 2 



