1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
431 
As for the permutations which are called reciprocal they are, 
exactly those whose existence we have seen noted by Rothe, and 
called by him “verw'andte Permutationen.” Jacobi’s definition, 
however, presents them in a slightly different light, the property 
involved in it being readily deducible from Rothe’s. The latter’s 
illustrative example was, as may be seen on looking back, 
3, 8, 5, 10, 9, 4, 6, 1, 7, 2 
8, 10, 1, 6, 3, 7, 9, 2, 5, 4 Bj . 
Now the performance of either A on B or B on A* gives rise to 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
the original arrangement : consequently A and B satisfy Jacobi’s 
definition. The proposition (h) is also Rothe’s. 
After these propositions, as already intimated, the subject of other 
rules of signs is taken up, the first rule considered being Cramer’s. 
Since in the product of differences corresponding to any permu- 
tation every factor in which an index is preceded by a smaller 
index would require the sign-factor — 1 to be annexed to it in order 
that the said product might be transformed into the original 
product of differences, it is clear that the determination of the class 
to which the permutation belongs is reduced to counting the number 
of such inversions. But the pairs of indices in the product of 
differences corresponding to the given permutation are exactly the 
pairs of indices to be examined in applying Cramer’s rule. The 
identity, of the two rules is thus apparent. (hi. 33) 
To the demonstration Jacobi adds “ quam regulam olim cel. 
Cramer dedit ill. Laplace demonstravit.” The last assertion is 
notable for two reasons : first, because the rule like Jacobi’s own 
is incapable of proof being a definition, postulate, or convention 
according to the mode in which it is expressed : secondly, because 
an examination of Laplace’s memoir shows that there is no ground 
for the statement. The fitness of the rule for the determination of 
the signs of the numerators and denominators of the unknowns in 
a set of simultaneous linear equations may of course be demonstrated, 
and perhaps this was in Jacobi’s mind, but prior to the, statement 
the abstract subject of permutations had alone been discussed. 
* In the compounding of reciprocal permutations the order is immaterial. 
This is the exception hinted at in ( d ). 
