928 Proceedings of Royal Society of Edinburgh. [sess. 
of what is called the “inverse 
determinant 
[ 1 , 1 ] [ 1 , 2 ] .... [ 1 ,»] 
[ 2 , 1 ] [ 2 , 2 ] .... [2 ,«] 
system, that is to say, the 
[n, 1] [n , 2] . . . . [n , n] 
where pqs] is the cofactor of (1 , 1) in 1 1 , 2 , . . . , n\. Cauchy’s 
theorem regarding the whole determinant is first proved, and then, 
instead of Jacobi’s more general theorem, there is established a 
theorem said to include Jacobi’s, viz. — 
\i + 1 , i + 1 ] 
[i + l,i + 2] 
.... p + 1 , n\ 
p -{- 2 , a + 1 ] 
\i + 2 , i + 2] 
.... p + 2 , n\ 
[«,* + !] 
[n , i + 2] 
[n,n] 
p+i + M'+Z+l] [i+j + l,i+j+ 2] ... [i+j+l,n] 
[i +j + 2 , i +j + 1] [i +j + 2 , i +j + 2] . . . [i +j + 2 , n] 
|l, 2,. ..,n\l 
1 1 , 2 , 
[n,i+j+ 1] [«,^+i + 2] ...\n,ri\ 
That it does include Jacobi’s is at once seen on putting i+j+ l=n, 
when we have 
p + 1 P + 1 ] p + 1 ,i + 2] . . 
p + 2 , i t 1 ] p' + 2 ji + 2] . . 
,. pH-1, rz] 
.. p+2,w] 
\n,i+ 1] \n,i + 2] 
.. [ 5 n,n ] 
[»,«]■! 1,2, ... ,n I” ‘- 1 - 
1 1 , 2 , . . . , » I’*-'- 1 . | 1,2, 
It is equally true, however, that by a double use of Jacobi’s theorem 
Spottiswoode’s follows immediately. 
The next section (§7) deals with the differentiation of a deter- 
minant, and with an application of the same by Malmsten to find 
the n ih particular integral of a certain differential equation when 
n- 1 particular intergrals are already known. 
The eighth section concerns the solution of what is called a 
“redundant system” of linear equations, that is to say, a system 
of m equations in n unknowns where m>n. Theorem XII. is the 
result obtained therefrom, and this being applied to the method 
of least squares, the last formulated result, Theorem XIII., is 
reached. 
• f +3 
.'I 
n- if 
