220 



DR THOMAS MUIE ON THE 



( «11 «12 «13 

 a 21 #22 a 28 

 a 31 a 32 a 33 



us specified in the preceding paragraph.* 



(9) It will have been observed that in the construction of the substitution use is 

 made of the peculiar matrix 



( a h + v ff — fx • - • ) 

 h-v b f+\ . . 

 g + fx f-\ c 



In fact this matrix is the material out of which the substitution is built up, just as the 

 matrix of a unit-axial skew determinant is the material for the formation of an 

 orthogonal substitution. And as in dealing with the problem of orthogonal substitution 

 Cayley was thus led to study the properties of skew determinants, we cannot do better 

 than follow his example and examine into the nature of the determinant of the analogous 

 matrix employed in the more general problem which we have just been dealing with. 



This more general determinant is seen to be obtainable by the superposition, so to 

 speak, of the zero-axial skew determinant 



X 



-X 



upon the axisymmetric determinant 



a h 



9 



h b 



f 



9 f 



c 



And as by changing the signs of all the Greek letters the value of the determinant is 

 unaltered, it follows that in the final development of the determinant there can be 

 no terms involving the product of an odd number of these letters. Further, as each 

 term of the final development of the determinant must be of the n th degree in the letters 

 which it contains, it follows that the said terms may be classified as follows : — 



1. terms of the w th degree in Italic letters 1 



and of the th Greek . . . J ' 



2. terms of the (n — 2) th degree in Italic letters | 



and of the 2 nd Greek letters J ' 



or, for shortness' sake, let us say terms of the degree n + 0, n— 2 + 2, n — 4 + 4,... 



* See also the third line of the proof that D -1 a'a~'D = a _1 a' in the footnote to § 6. 



