56 



DR THOMAS MUIR ON 





1.3.5.7.9 - 3(1.3.5.7), 



i.e., 



1.3.5.7(9 - 3), 



i.e., 



35 x 18; 



and this is exactly the number obtained from our development. 



(8) The only matters remaining now for consideration are those which concern the 

 signs of the terms thus obtained, it being necessary for our purpose (1) to establish the 

 fact that the eighteen signs of any product of the form 



15 



17 



18 



. | 46 49 U 



25 



27 



28 



69 U 



35 



37 



38 



9t 



are either all right or all wrong, and (2) to formulate a rule for distinguishing 

 between products of these two kinds, so that the sign + may be prefixed to the one and 

 — to the other. 



Now if no sign precede the product, the sign of any of the eighteen terms is 

 determinable from the sign of the portion of the term which comes from the deter- 

 minant factor and the sign of the portion which comes from the Pfaffian factor ; 

 and the former being dependent upon the number of inverted-pairs in the series of 

 column-numbers specifying the elements of the first part of the term, and the latter 

 upon the number of inverted-pairs in the series of frame-line numbers specifying the 

 elements of the second part of the term, it is clear that if <r be the sum of the said two 

 numbers of inverted pairs the sign of the complete term will be ( — Y. On the other 

 hand, the sign which it ought to bear as a term of the parent Pfaffian is fixed by the 

 number v of inverted-pairs in the series of integers specifying the frame-lines of all 

 the elements composing it. What is wanted, therefore, is a comparison of this number 

 v with a- ; and, if we can show that v — a- is constant for all the eighteen terms, it will 

 follow, of course, that the eighteen signs are either all right or all wrong. For 

 example, in the product 



15 



17 



18 



.46 49 At 



25 



27 



28 



69 6* 



35 



37 



38 



9* 



— 17.25.38 is a term of the determinant and — 49.6t a term of the Pfaffian, the sign 



— in the one case being fixed by the number of inverted-pairs in 758 and in the 

 other by the number in 496£, whereas the sign of the resulting term 17. 25. 38. 49. 6£ of 

 the Pfaffian is fixed by the number of inverted-pairs in 172538496^. 



For ease in making the necessary comparison let us use 



I(a/3y....) 



to stand for the number of inversions found in the pairs of integers obtainable by 

 placing each integer in front of those which follow it, and 



I(a/3y .... ; a'yS'y' . . . . ) 



