Note on Clebsch's Theorem. 
395 
4. With the same notation 
'''Tx 
^dy ^oy "^dy 
ciVj 'bv^ Tiv. Tiv, ^ 
"■liz ^liz "^z 
(r) 
Differentiating (a) with respect to one of the independent variables, 
say X, we have 
/ I '^x 
= 0, 
and since, by (/3), the second sum here is equal to 0, so also must the 
first sum. 
5. Solving now the equations (y) for the w's we obtain at once 
Clebsch's result : in fact (y) may be viewed as simply another way of 
saying that the 2^'s are proportional to 
6. The next requirement is to explain the cause of the efficacy of 
Clebsch's irrelevant-looking determinant of the (7z-i-2)th order. In doing 
this we need not confine ourselves to elements that are differential- 
coefficients : the determinant to be bordered may be any determinant 
whatever, anything peculiar being introduced along with the border, and 
indeed with only part of that. Let us take, for example, the general 
determinant of the fourth order \a^^c^d^. The added row of four 
elements then is 
^3> ^4 
where the e's again are any quantities whatever: but the added colunui is 
0, 0, 0, m^d^-\- ... +77^4(^4, mi<3i + ... -i-m^c^: 
and the new theorem to be established is 
a^ a^ «3 a^ 
b. b^ 
^1 ^2 ^4 
d^ d^^ d^ ^md 
61 ^2 63 e^ %7ne 
... + \d.^e,l\B^m,\, 
where D,. is the cof actor of d, in [a^b^c.d^l. 
