1888-89.] Dr T. Muir on the Theory of Determinants. 433 
cycle be a v a 2 , a 3 , ... , a r , it is clear that to change a Y into a 2 , a 2 into 
a 3 j &c., has the same effect as to interchange a x and a 2 , then a x and 
a 3 , then a 1 and and so on, the final interchange being that of a x 
and a r ; and there are in all r - 1 interchanges. This being proved, 
the final step is taken as in Cauchy’s Note of 8th March, (iii. 34) 
This rule of Cauchy’s Jacobi deservedly characterises as beautiful. 
It is important, however, to take note that it possesses the other 
quality of usefulness in as marked a degree ; and such being the 
case one is surprised to find that it has not received the attention 
which was its due. Any reader who will make a comparison of it 
and Cramer’s by actual application of them to a number of examples 
will soon find that Cramer’s is more lengthy and requires more care 
to be given to it to avoid errors.* 
The preliminary subject of permutations having been thus dealt 
with, determinants are taken up. In the first section regarding them 
there is little noteworthy. Cauchy’s word “terme” is supplanted by 
the fitter word element , and term (“terminus”) is put to a more 
appropriate use ; that is to say, a® is called an element of the 
determinant 2 ± aa\a" 2 . . . and ’a k a a" k aff a term. 
Further, the word degree is employed in place of Cauchy’s more 
suitable word order , “ipsum R dicam determinans n + l ti gradus .” 
A section of two pages is given to considering the effect produced 
upon the aggregate of terms by the vanishing of certain of the 
elements. The propositions enunciated, with the exception of one 
made use of at an earlier date by Scherk, are as follows (pp. 291, 
292):— 
“ I. Quoties pro indicis k valoribus 0, 1, 2, . . . , m - 1 
evanescant elementa , ... . , determinans 
"L±aa x a a^ 
1 A n 
abire in productum a duobus determinantibus 
2 ± aal .... «<::;> . 2 ± a ( y:^ (XIV. 6) 
* The best way perhaps of applying Cauchy’s rule is to write the primitive 
permutation, 128456789 say, above the given permutation, 683192457 say, 
draw the pen through 1 and the figure below it, seek 6 in the upper line and 
draw the pen through it and the figure below it, and so on, marking down 1 on 
the completion of every cycle. 
VOL. XVI. 16/11/89 2 B 
