524 DR THOMAS MUIR ON VANISHING AGGREGATES 



With the help of this notation the theorem of § 17 can be enunciated in an exceed- 

 ingly simple way, viz., 



"p.2, ...,«- 2; n-l, n, ..., 2n - 2~| 

 L n-l ,«,..., 2n-2J 



hen 



it — 1 , n , . . . , 2n — 2 

 n - 1 , n , . . . , 2n - 2 



is axisymmetric." 



(19) If to n — 2 — a given columns of n elements each there be appended the Cj tt , 

 c 2 th , • • • , c a tt awc£ r® columns of a given determinant of the n th order, and from the 

 resulting array the r th row be deleted, there being thus produced a square array of 

 the (n — l) th order the determinant of which is D r , then the aggregate 



r=n 



will vanish tvhen the given determinant is axisymmetric. 



By way of proof it is only necessary to note in the first place, that if we increase 

 the n — 2 — a columns of the non-quadrate array by repeating a of the columns which 

 occur in the quadrate array, the former array will consist of n — 2 columns, and the 

 theorem of the preceding paragraph will be applicable : and in the second place, that 

 when this application is made, the result obtained is the more general theorem just 

 enunciated. 



Of course the number of terms in the aggregate will not now be n but n — a, as 

 D r will have two identical columns, and therefore will vanish when r equals any one 

 of the series c x ,c 2 , ... , c a . 



Further, the number of identities connected with the two arrays is now not 1 by 



(20) Although the theorem of § 18 in the form there given is so readily deduced 

 from that of § 17, it being possible to view it as the particular case of the latter where 

 a columns are common to the two given arrays, the state of matters is very different 

 when we seek, as has been done at the end of § 17, to obtain the development of 

 2/ — l) r_r D r in terms of the minors of the (n — 2 — a) th order formable from A. The 

 theorem of § 18 is then unmistakably the more general, that of § 17 being merely the 

 case where a = 0. The law of the development, too, is then, not by any means lying 

 so close to the surfai-e, and almost requires for its proper expression the use of a series 

 of new integral functions connected with the theorem referred to in the third sentence 

 of § 1. 



(21) There has first to be recalled the well-known theorem of Kronecker, some- 

 times insufficiently described as giving a linear relation between minors of an axi- 

 symmetric determinant, but which really asserts that certain aggregates of n+ 1 minors 

 of the n th order belonging to a determinant of the (2n) th order vanish when the latter 

 determinant is axisymmetric. For example, when n — 3, it asserts that 





