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II. 
0^ A NEW METHOD OF OBTAINING THE CONDITIONS 
FULFILLED WHEN THE HAEMONIC DETERMINANT 
EQUATION HAS EQUAL EOOTS. By FEANCIS A. 
TAELETON, LL.D., F.T.C.D. 
[Bead November 14, 1887.] 
The harmonic determinant possesses a peculiar interest, botli on ac- 
count of tlie importance of tlie physical problems with which it is 
connected, and also because it is associated with the names of some 
pre-eminently great mathematicians. 
Lagrange, in the Mecaniqiie Analytique, shows that the small oscil- 
lations of a system whose position is determined by n independent 
variables depend on the solution of an equation of the n^^ degree. 
This equation in its most general shape is of the form 
Pn - ^fl2 P22 - ^/23 . • •P2n - ^An 
= 0. 
Pin - ^fln Pnn " 
The determinant which enters into this equation is called the har- 
monic determinant, as each root of the equation corresponds to an 
oscillation which all the variables of the system complete in the same 
time. 
Lagrange appears to have been but slightly acquainted with the 
properties of this determinant, and, as is now well known to mathe- 
maticians, was mistaken in reference to the consequences which result 
when the equation above has equal roots. Laplace fell into the same 
mistake as Lagrange, and Dr. Eouth of Cambridge has the honour of 
being the first to point out the error of the great French mathe- 
maticians. 
It would, however, have been scarcely possible for this error to 
have remained much longer undetected. Looked at from the physical 
