Tarleton — The Harmonic Determinant. 
15 
In general, if the equation A = 0 have m roots equal to X, all tlie 
first minors of A, of An, of An 22, and so on up to and including 
A1132 . • . (m-2)(m-2)5 niust vauisli. 
It is well to observe that equation (3) follows immediately from a 
more general theorem given in Salmon's Higher Algebra, and which, 
for any determinant, is expressed, according to our notation, in the 
form 
AnA22 - Ai2A2i = AAn22. (4) 
When the determinant A is symmetrical, and likewise zero, this equa- 
tion becomes the same as (3). 
Mr. "Williamson has recently pointed 'out that, from equation (4), 
it follows that in the case of the harmonic determinant, when An 
vanishes, A and An 23 take opposite signs. Hence can be obtained an 
easy and beautiful proof of the reality of the roots of the equation 
A = 0. This proof is to be found in Salmon's Higher Algebra, but is 
there rendered somewhat difficult by the generality of the investiga- 
tions with which a part of it is intermixed. 
