of Edinburgh , Session 1871 - 72 . 
783 
the first of the above equations gives, by the help of the second, 
after the elimination of the displacements 
1 
- 2 1 
&c. 
1 
1 
= 0 . 
1-2 1 
1 l - 2 
This is a particular case of the determinant, 
p q r s . 
z p q r . 
y « p q . 
2 p q 
y z p 
which, equated to zero, gives the result of elimination of 0 between 
the equations 
p + q6 + + . + slT -1 = 0, 
0 " - 1 = 0 . 
Its factors are obviously to be found by substituting in succession 
the several m th roots of unity in the expression 
p + qO -f .+ zb m ~ x . 
The form of its minors, on which depends the solution of the pen¬ 
dulum question, follows easily from these properties; and from 
them we in turn easily obtain the value of the same determinant 
when bordered, as it will be in the pendulum case if the series of 
magnets be finite and not closed. The question forms a very in¬ 
teresting illustration of the linear propagation of disturbances in a 
medium consisting of discrete, massive, particles—when only con 
tiguous ones act on one another. For, if we put 
d 
-r-. a -J- 
D = e d y > 
