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XL— A Pfaffian Identity, and related Vanishing Aggregates of Determinant 



Minors. By Thomas Muir, LL.D. 



(MS. received February 26, 1906. Issued separately August 16, 1906.) 



(1) An essential part of Pfaff's method of solving an ordinary differential equation 

 in 2m variables consists in obtaining what he calls his auxiliary equations. If the given 



equation be 



Ada + Bdb + Cdc + Ede + Yd/ + Gog = 



the auxiliary equations are 



0= da 



(CBE) (CFG) - (CBF) (CEG) + (CBG) (CEF) 

 db 



"■" (CAE) (CFG) - (CAF) (CEG) + (CAG) (CEF) ' 



— ? 



if the given equation be 



Ada + Bdb + Gdc + Ede + Yd/ + Gdg + H3A + Idi = 



the auxiliary equations are 



_ da 3c 



^ + 6 ~ •••' 

 where 



21 = (BCE) (BFG) (BHI) - (BCE) (BFH) (BGI) + (BCE)(BFI) (BGH) 



- (BCF) (BEG) (BHI) + (BCF) (BEH) (BGI) - (BCF) (BEI) (BGH) 

 + (BCG) (BEF) (BHI) - (BCG) (BEH) (BFI) + (BCG) (BEI) (BFH) 



- (BCH) (BEF) (BGI) + (BCH) (BEG) (BFI) - (BCH) (BEI) (BFG) 

 + (BCI)(BEF)(BGH) - (BCI) (BEG) (BFH) + (BCI) (BEH) (BFG) , 



and so on, it being explained that 



/T} _„, B9C-C3B E3B-B3E C3E-E3C 

 < BCL > = — fc— + — dc— + — 37T- ■ 



A law for the formation of the denominators occurring in those auxiliary equations 

 Pfaff himself gave, and through the interest taken in it by Jacobi and Cayley it has 

 been for more than half a century well known. Pfaff, however, also obtained his 

 auxiliary equations in a second form, with denominators quite unlike those of the other 

 in appearance ; and though he again carefully enunciated the law of formation, a very 

 different fate in this case supervened, much to the detriment of the theory of Pfaffians 

 and determinants. 



TRANS. ROY. SOC. EDIN., VOL. XIV. PART II. (NO. 11). 43 



