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XXIV. — On a Development of a Determinant of the mn"' Order, 

 By Thomas Mum, LL.D. 



(Read 20th March 1899.) 



(1) The theorem in question concerns the finding of an expression for a determinant 

 of composite order in terms of compound determinants or permanents of lower order ; 

 or, more definitely, the expression of a determinant of the mn th order as an aggregate of 

 determinants or permanents of the m th order, each of whose elements is a determinant of 

 the n til order. 



(2) Takiug the simplest possible case, viz., where the given determinant is of the 4th 

 order, we have by Laplace's expansion-theorem 



a v 



«, 



a s 



a 4 



h 



h 



h 



h 



h 



Co 



c z 



c i 



d x 



d, 



d 8 



d i 



I a A I ' I C A I - I "i 6 ' 2 1 ' 1 h A I + I *A I ■ I h G i I 



+ I \h I ' I *A I - I W-2 I ■ I «3 C 4 I 



+ I c A I ' I a A I • 



Now the sum of the first and last terms here is clearly equal to the permanent 



j I a ih I 

 I I «A I 



the sum of the second and fifth equal to 



+ 



_ I I a i C 2 I 



" I I b A I 



and the sum of the third and fourth equal to 



C A\ 



h d i 



I <h d 2 I I a A I I 



1 h ( --i I 1 h c i I I ; 



so that we have 



+ + 



+ -i- 



"A* A = 1 1 «A I I cA 1 1 - 1 1 «v» I 1 ^ 3 ^4 1 1 + | ! <*A I i h c ± 1 1 



VOL. XXXIX. PART III. (NO. 24.) 



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