54 



DR THOMAS MUIR ON 



/ 



- 9 - 



- % - - - 



m n 



and if we next decided on taking Jc, which is in the 3rd and 5th frame-lines, we could 

 not thereafter take anything else from these lines — that is to say, we could not take j or 

 I from the 3rd frame-line, or m or o from the 5th. We should thus be left with 



ahn 



as a term; and the number of inverted-pairs in the series 12 35 46 made up of the 

 place-numbers of the chosen elements being 1 , the sign would be negative. 



(6) It is of interest to note, in passing, that the term composed of the 1st, 3rd, 5th, 



. . . elements in the hypotenuse of the semiquadrate array is always + , whatever the 



order of the Pfaffian may be, because the number of inverted-pairs in 12 34 56 . . . . 



is zero. Also, that the same is true of the term composed of the elements lying on the 



line which bisects the hypotenuse at right angles, because the series then to be 



considered is 



1,2k, 2,2fi-l, 3,2rc-2, n-\,n; 



and it is manifest that none of the pairs beginning with 1, 2, 3, . . . , n— 1 can be in- 

 verted, and that, while those beginning with 2n, 2n—l,..., are all of them inverted, 

 the number is in each case even. 



(7) With these preliminaries before us, let us now consider a fourth case of the 

 theorem sought to be established, say the case where the Pfaffian is of the 5th order and 

 the zero elements are in the places 12, 13, 23 — i.e., the Pfaffian 



I . . 14 15 16 17 18 19 It 



. 24 25 26 27 28 29 2t 



34 35 36 37 38 39 3t 



45 46 47 48 49 At 



56 57 58 59 5t 



67 68 69 U 



78 79 It 



89 8* 



9t 



