Tarleton — The Harmonic Determinant, 
11 
standpoint, Lagrange's solution in tlie case of equal roots was readily 
seen to be incorrect, by taking tbe extremely simple case of a particle 
disturbed from tbe lowest point of tbe internal surface of a smootb 
sphere ; whilst on the purely mathematical side the increasing know- 
ledge of Higher Algebra soon led to a complete acquaintance with the 
properties of the harmonic determinant. 
It is strange that, nevertheless, no simple investigation of the con- 
ditions fulfilled when two or more roots of the equation in \ become 
equal has been given, so far as I know. 
The subject is discussed in Salmon's Higher Algebra, as well as in 
Thomson and Tait's I^Tatural Philosophy ; but in each of these works, 
especially the latter, it presents considerable difficulty to the student. 
Burnside and Panton on this matter have not, I believe, added any- 
thing to what is to be found in Salmon's Higher Algebra. In "William- 
son and Tarleton's Dynamics, a mode of investigation obscurely indicated 
by Thomson and Tait is fully developed ; but this method, though not 
very difficult, is highly artificial in its character, and was adopted in 
the Dynamics, merely on account of the length and difficulty of the 
other known methods of investigation. 
A special case of the more general problem occurs in finding the 
conditions fulfilled by the general equation of the second degree when 
it represents a surface of revolution. The mode in which this question 
is treated in Salmon's great work throws but little light on the general 
problem. A less special mode of investigation can easily be given as 
follows : — 
The equation of a surface of the second degree, referred to its axes, 
is of the form 
A^' + Brf + (7^ - K. 
Eefore transformation, the origin being the centre, it was of the form 
If we put F = 1^ + + C^, the discriminant, A, of U- XV is 
{A-\){£-X){C-X). 
If the surface be one of revolution, A = and the equation A = 0 
has equal roots, the value of the double root being A. 
In this case IT- AV= {C - A)t,^, and therefore 17- AV is a 
