Tarleton — The Harmonic Determinant. 
13 
variables their differential coefficients with respect to the time, we 
obtain an expression for the vis viva of the system. Hence V is 
always positive, and may therefore, by linear transformation, be ex- 
pressed as the sum of the squares of the n variables. 
If U\ie transformed into a function of these same variables, the 
harmonic determinant becomes 
Pn - A i?i2 pm 
Pn P22-^ P23 P2n 
Pin P2n Pin 
This determinant we shall call A. 
If we erase the first row and first column of A, we get a new 
determinant which may be called An. In like manner, A22 may be 
used to signify the determinant got by erasing the second row and 
second column of A, and A12 that obtained by erasing the first row 
and second column, and so on. 
Let us now consider the determinant 
PlX - Ai i?i3 i?13 . . . Pin 
P12 P22 - A2 i?23 . • • P2n 
hn P2n Pzn 
Pr, 
which we may call A', in which Ai, A2, &c., are functions of X. 
We have, then, 
d^' _d^' dA^ ^ d^' dA^ ^ 
dX dAi d\ dA^ dX 
If we now suppose 
Ai = Ag = A3 = &c. = A, 
the equation above becomes 
dA 
= - (An + A22 + &c. + A„„)- 
(1) 
Hence, if A be a double root of the equation A = 0, we must have, for 
this value of A, 
An + A22 + &c. + A„, = 0. (2) 
