595 
of Edinburgh^ Session 1865 - 66 . 
i+‘ I I I t‘ 
1^1-2 ^l-2-3 1-2-34 
+ &C. 
and it is shown that the fundamental recurring functions of any 
higher order, as the are obtained by taking each term of this 
development. 
When each alternate term of the series for is taken, we obtain 
a function which is equal to its own second derivative ; of this func- 
tion there are two varieties, according as the terms contain the even 
or the odd powers of the primary. If the value of the primary be 
represented by abscissse, and the corresponding values of the func- 
tion be indicated by ordinates, we obtain two curved lines, one of 
which is the catenary, and the other, a line which may be called 
the companion to the catenary; these two lines do not meet each 
other. 
If we take each third term of the development of eq we obtain 
recurring functions of the third order; of these there are three va- 
rieties, according to the term with which we begin. When the 
values of these three functions are represented by ordinates, there 
result three curved lines which intersect each other, and it is shown 
that their intersections take place on ordinates at equal distances 
from each other, the lines being, as it were, plaited upon each 
other. As the value of the primary is augmented, the interval be- 
tween the curves, as measured on an ordinate, generally diminishes, 
and the three lines soon become so close as to be undistinguishable 
in a drawing of ordinary size. For negative values of the abscissae, 
the curves separate more and more from each other. The distance 
between the ordinates, on which these intersections take place, is 
an important feature of the ternary functions ; it bears a certain 
relation to the circumference of a circle of which the radius is equal 
to the linear unit, and is susceptible of very easy computation. 
A very remarkable property of the lines representing these ternary 
functions is this, that if an equilateral triangle be placed in a plane 
perpendicular to the plane of the paper, and passing through one 
of the ordinates in such a way as that the three corners of the trigon 
may have the points of the three curves for their projections ; 
and if the ordinate be supposed to be displaced along the line of 
abscissae at a uniform rate, the trigon will turn round also with a 
uniform velocity, and its side decreases or increases in continued 
