52 



DR THOMAS MUIR ON 



We thus have as a final result 



<*6 



a 7 



a s 



= ~ | «2 & 4 | • 



I e 6 



e 7 e s 



^6 



h 



h 







fl /s 



C 6 



c 7 



c s 







9s 



** 



d 7 



d s 









e 6 



e 7 



A 



e s 

 /s 

 <7s 









- I a-l* i 



"4 °5 °6 



where the first factors of the terms on the right are the fifteen (i.e., C 6i2 ) two-lined 

 determinants forma ble from 



a s a 4 a 5 a G ct 7 cc s 

 \ ^4 h & <i h h 



and their cofactors are the fifteen (i.e., 5 + 4 + 3 + 2 + 1) principal minors of the Pfaffian 



C 5 



C (3 



C 7 



c s 



h 



d. 



d 7 



d s 





e 6 



e 7 



e s 







A 



fs 

 9s 



the first determinant | a 3 6 4 [ going along with the complementary minor of the first 

 element c 4 of the Pfaffian, the second determinant | a 3 b 5 | going along with the comple- 

 mentary minor of the second element c 5 of the Pfaifian, and so on in every case. 



(4) For the full investigation of the general theorem thus shadowed forth, neither 

 of the definitions here employed is well suited : what is needed is a definition prescrib- 

 ing the mode of formation of the terms from the elements and the mode of determining 

 the sign of each term — a definition, that is to say, similar to that ordinarily used for a 

 determinant. The following will be found to satisfy these requirements : — 



If n(2n— 1) elements be each numbered by a 'pair of integers in order of magni- 

 tude, and be arranged in semiquadrate form, thus — 



12 13 14 

 23 24 



l,2n 



2,2n 



2n-l,2n 



and all possible terms be taken which are products of n elements whose united place- 

 numbers include all the integers from 1 to 2n, the sign of each term being taken + or 

 — according as the number of inverted-pairs in the series of integers specifying the 

 term is even or odd ; then, the function which is the aggregate of these terms is called 

 a Pfaffian of the n th order, and is denoted by the semiquadrate collection of elements 

 bounded by two straight lines, a shorter on the left and a longer on the right. 



