b+c+d=B
c+d+e=C
d+e+a=D
e+a+b=B
A+B+C+D+E=S.
We may solve and obtain:
b=B+E-12S
c=C+A-12S
d=D+B-12S
e=E+C-12S
The writer was surprised to find, however, that while in some cases these equations were readily solved, in others they were impossible of solution. My friend, Mr. Carl G. Barth, when the matter was referred to him, soon developed the fact that the number of elements of a cycle which may be observed together is subject to a mathematical law, which is expressed by him as follows:
The number of successive elements observed together must be prime to the total number of elements in the cycle.
Namely, the number of elements in any set must contain no factors; that is, must be divisible by no numbers which are contained in the total number of elements. The following table is, therefore, calculated by Mr. Barth showing how many operations may be observed together in various cases. The last column gives the number of observations in a set which will lead to the determination of the results with the minimum of labor.
