1888-89.] Dr T Muir on the Theory of Determinants. 405 
bases in order to remove the difficulty. At any rate the difficulty 
is removed ; for the number of terms in the period being 5 the 
index-numbers 4 and 5 become changeable into - 1 and 0, and thus 
we can have 
| | = j I > 
= I $_1?>0 C 1^2 I 1 
— a determinant in which the index-numbers proceed by the 
common difference 1, and which is obtainable from | | by 
diminishing each index-number by 2. Sylvester’s form of the 
result thus is 
£ •JS 2 («M).^PD(0aM)| = £_ 2 (0 abed).* 
Following this comes the application to simultaneous linear 
equations, or as they are called “ equations of coexistence.” The 
system is represented by the typical equation 
a,x + b r y + c r z+ . . . + l r t = 0 , 
in which r can take up all integer values from - oo to + oo , there 
being really, however, only n equations, because of the periodicity 
imposed on the arguments of the bases. One so-called “leading 
theorem” is given in regard to the system, its subject being the 
derivation of an equation linear in x, y, z, . . ., t by a combination 
of the equations of the system. The theorem is enunciated as 
follows (p. 40) : — 
“Take /, g, . . ., Jc as the arbitrary bases of new and ab- 
solutely independent but periodic arguments, having the same 
* It is rather curious that Sylvester overlooks the fact that the legitimate 
equatemeut of two zeta-ic products implies an identity altogether independent 
of the existence of zeta-ic multiplication. Thus, the identity just discussed is 
essentially the same as the identity 
a 
a 2 
a? 
a 4 
a 
a? 
a 4 
a 5 
b 
6 2 
6 3 
6 4 
x {ah + ac + ad + be + bd + cd ) = 
b 
b 2 
6 4 
b 5 
c 
c 2 
c 3 
c 4 
c 
c 2 
c 4 
c 5 
d 
d? 
d 3 
d x 
d 
d? 
d i 
d 5 
where the index-number denotes a power and the multiplication is performed 
in accordance with the ordinary algebraic laws. From this point of view the 
above quoted proposition of Sylvester’s involves an important theorem re- 
garding the special determinants afterwards known by the name of alternants. 
