155 
of Edinburgh, Session 1866-67. 
that is to say, the fourth derivative is quadruple of the function 
with the sign changed. This brings us to a new class of functions 
in reality of the eighth order, and bearing to the quaternary func- 
tions the same kind of relation which the circular functions, cosine 
and sine , bear to the catenarian ones. 
If we write instead of x, z we have 
E> ! - GEF = 1 - Ai + 7T§ ~ etc - ’ 
in which the signs are alternately + and — ; denoting this func- 
tion by the character <£> z, and its derivatives by — <7>z, — <7>z, 
- <i) 2 , we have four functions, viz., 
II 
St 
<$> 
1 
z* 
.“74 
+ 
z 
778 
— etc. , 
<j>z = 
z 
1 
z 5 
• .5 
+ 
z 9 
...y 
— etc. , 
II 
<$> 
z 2 
1.2 
z 6 
...6 
+ 
z 10 
...10 
— etc. , 
II 
<s> 
z 3 
z 7 
+ 
z' 1 
— etc. , 
1.2.3 
...7 
....11 
which are almost the counterparts of the quaternary functions, and 
of which the values may be computed with great ease. And it is 
noteworthy that by putting z = these functions reproduce 
the quaternary ones. 
The determination of the values of the coefficients in the equa- 
tions of the curve, is thus reduced to the numerical solution of the 
two equations 
<7> z = — 1, and <4> z = +1, 
and situated between these there is this, viz., 
<$> = °, 
the roots of which are the odd multiples of . 
The results of the calculation, as applied to the case of an elastic 
rod held firmly by one end, were stated to be, that the funda- 
mental or slowest vibration, and the next one, have their periodic 
times in the ratio of 6‘2557 to 1, or very nearly as 25 to 4. And 
that the node in the second vibration is at the distance ‘78345 
