Determinants and Pfaffians. 



239 



As we know otherwise* that the vakie of the determinant is 



la (3 \ \ a yl |y3 y!, 



an expression which is symmetrical with respect to the interchange 



fab c 

 ^ a /3 y 



we arrive at the following curious series of identities 



or 



a b c- 2 

 a li y 



62 + C-2 —ab—y —ac + ^ 



— ba + y c^ + a^ —be — a = 

 -ca-l3 -cb + u a^ + b- [ 



« ^ 7 _ 



b^ + c^ — ab —ac I o 



— ba c^ + a^ —be (3 

 — ca —eb a^ + b' y 



fie — by a a 

 ya — Ca (3 b 



ab—aji y c 



/32 + y2 _^a — C —ya-\-b 



—I3a + C y^ + a^ —f3y — a 



— ya — b —y^+a a^ + fi^ 



a be 



* Performing the operations : 

 b 



, h . c 



row J +— rowj + -row 3. 



coll +— C0I2 + — coL, 

 a a 



and denoting 



^c — by. yd — ta, ab — aji by A, B, C 



so that aA + 6S + cC = o = aA -\- fiB -\- yC we obtain 



B C 



B 



c--\-a'- — be — « 



-bc-\-a a--\-b- 1, 



which 



= A j 2bcBC + C'(c-+a") + Br-(a-+b-) \ 

 a- '■ ' 



= l^iE±^y + c- + B^ 





= .4- + B- + C-. 



