420 



Dr. T. Muir on an overlooked Discoverer 



sixteen terms thus obtained. In fact the sixteen terms may 

 be written in order in four rows of four each ; and these 

 being viewed as four columns of four each, and the columns 

 summed, we obtain the right-hand member of the identity. 



It is then shown that this is a case of a general theorem in 

 which the factors of each product are determinants of the 

 (n-\-l)th and mth orders, the said theorem being accurately 

 and succinctly stated by means of a suitable notation. 



Even here, however, the matter does not end ; for by a 

 quite natural step Schweins passes on to a much more widely 

 general theorem, in which the factors on the one side are 

 determinants of the (p + s + q)th and (k + p)th orders, while 

 those on the other are determinants of the (p + s)th and 

 (q + h +p)th. orders. 



The conclusion of the chapter is occupied with the state- 

 ment of a considerable number of interesting special cases of 

 the latter theorem. 



6. There can be no doubt that almost every detail of this 

 chapter was new at the time of publication; and it is therefore 

 of some importance that the nature of its contents be properly 

 understood. This will be most readily attained if we present 

 them shortly in the light in which nowadays they would 

 naturally be viewed. 



In the case of the special identity with which the chapter 

 opens, a writer of the present time would only require to 

 direct attention to the determinant 



a x a 2 a 3 a A 



h h h h . 



C\ c 2 c z c± . 



d x d 2 d 3 d 4 d- d 6 d 1 



e 1 e 2 e 2 e± e 5 e 6 e 7 



f\ J 2 ./3 fi tb A fi 



9\ 92 9s 9± 9b 9e 9i 



whose non-zero elements form a quadrate gnomon of the 

 dimensions 4, 7, 4 ; and point out that this determinant is 

 the natural sum, according to Laplace's expansion-theorem, 

 of the left-hand member of the identity, and likewise of the 

 right-hand member ; the former being obtained by developing 

 the determinant in terms of the minors of the 4th order got 

 from the first four columns, and the latter by developing it in 

 the same way in terms of the minors of the 4th order got from 

 the last four rows. 



