( 511 ) 



XXII. — Vanishing Aggregates of Secondary Minors of a Persymmetric Determinant. 



By Thomas Mum, LL.D. 



(Read 19th May 1902.) 



(1) The persymmetric determinant of the n th order 



a i a 2 



a., a., 



a Q a, 



a. 



«n a n 



a, n 



being such that in every case the element in the place r , 5 is the same as the element 

 in the place r— 1 , s+ 1 , and therefore having only 2n — 1 independent elements, viz., 

 the elements a 1 , a 2 , .... , a 2m _i forming the first row and last column, is conveniently 

 denoted by 



P («l , «2) ••' » «3n-l) • 



As it is a special form of axisymmetric determinant, any known relation between 

 minors of the latter must of course hold in regard to the corresponding minors of the 

 former. Now in the case of the axisymmetric determinant 



12 3 4 5 

 12 3 4 5 



it was shown in April 1897 that there existed between three of its secondary minors 

 the relation 



1 3 fi I i 1 9 S I I 1 L S 



= 0; 



and it was pointed out how this and other similar theorems might be generalised.* A 

 few months later there was published in Italy a paper the object of which was to 

 establish the following theorem : — If in a 'persymmetric determinant the I st and x th 

 rows be deleted and the 1 st and s th columns, the secondary minor so obtained is equal 

 to the stem oftivo similar secondary minors, one got by deleting the 1 st and (r + l) th roivs 

 of the original and the 1 st and (s - l) th columns, the other by deleting the 1 st and 2 nd rows 

 and the (s — l) th and r th columns, f If we indicate by 



a, {3, y, . 



Z,v,0, ■ 



* Muir, Th., "The automorpliic linear transformation of a quadric," Trans. Roy. Soc. Edin., xxxix. [pp. 209-230] 

 p. 226. 



t Cazzaniga, Tito, " Relazioni fra i minori di un determinante di Hankel," Rendiconti del R. 1st. Lomb. di sc, 

 e lett, serie ii. vol. xxxi. (1898) pp. 610-614. 



TRANS. ROY. SOC. EDIN., VOL. XL. PART III. (NO. 22). 4 h 



1 



3 



5 



1 1 



2 



5 1 



1 



4 5 



1 



2 



4 



1 



3 



4 



1 



2 3 



