1907-8.] Dr Muir on Compound Determinants. 
197 
X. — The Theory of Compound Determinants in the Historical 
Order of its Development up to 1860. By Thomas Muir, LL.D. 
(MS. received August 20, 1907. Read November 4, 1907.) 
Determinants whose elements are themselves determinants made their 
appearance at a very early stage in the history of the subject, the first 
foreshadowing of them being contained in Lagrange’s “ equation identique 
et tres remarquable ” of 1773, namely, 
+ vCF + & v" ~ - vi'l" - b)%' = ( x y'z + yz'x + zx f y " - xzy - yxz - zyx ") 2 , 
where 
& y, £,...= y'z - y"z, z'x” - xy - xy\ . . . 
This, viewed as a result in determinants, is a case of Cauchy’s theorem of 
1812 regarding the adjugate, and the adjugate of course is an instance of 
the special form to which we have now come. Jacobi’s theorem regarding 
any minor of the adjugate has a like history and may be similarly 
classified. Passing from the case of the adjugate, where each element is a 
primary minor of the original determinant, Cauchy also considered the 
determinants of other “ systemes derives,” that is to say, the determinants 
whose elements are the secondary, ternary, . . . minors of the original, 
and gave the theorem that the product of the determinants of two 
“ complementary derived systems ” is a power of the original determinant, 
the index of the power being 
n(n- \)(n- 2) ... (n -p + l)/l.2.3 ...p, 
where n is the order of the original determinant and p the order of each 
element of one of the “ derived systems.” He also in the same memoir 
established the theorem that the determinant of any “ derived system ” of a 
product-determinant is equal to the product of the determinants of the 
corresponding “ derived systems ” of the two factors. 
Those are all the general results that fall to be noted prior to the middle 
of the nineteenth century ; and, as is readily seen, they all concern what 
at a later date came to be called the “compounds” of \a Xn \. With one 
exception they are due to Cauchy.* 
The fact has also to be recalled, however, that compound determinants 
* They are numbered xx., xxi., xli., xlii. in my History. 
