488 
PHYSICS: H. S. UHLER 
apparent regular sequence of the lines and the degree of accuracy with 
which their wave-lengths or their frequencies fit a smooth curve or 
satisfy an empirical equation. It is therefore evident that any formula 
proposed for band spectra must be subjected to very severe tests before 
it can be accepted as an expression of a law of nature. 
A very general law for all series spectra was proposed by Thiele 
in the year 1897. It was expressed by the equation 
X=/[(« + c)^], (1) 
where X denotes the wave-lengths of a line whose ordinal num^ber is 
and c is a constant for any one series. Thiele styled c the "phase'' 
of the series. The integer n is supposed to assume negative as well as 
positive values. Consequently a series must in general be composed 
of two groups of lines each of which would ordinarily be called a series. 
More precisely, the positive branch of each series must be accompanied 
by a negative branch of the same series, having the same head = 0) 
and the same ^taiF = oo) and being represented alternately by a 
line in each interval of the other branch. In the case of band spectra 
Thiele highly recommended the special form of function (I) which 
he employed with apparent success in his computations of the wave- 
lengths of the lines of the carbon band at X 5165 and which he wrote as 
X = Xo- 
The objects of the present paper are to outline a practical method 
for calculating c without assuming a special form of function (1) and to 
give briefly the results obtained by applying the new method to the 
X 5165 carbon band and the third cyanogen band at X 
It should be remarked in advance that it is not feasible to solve 
equation (2) for c when a sufficiently large number of wave-lengths 
are used to ensure a satisfactory degree of accuracy. The adjacent 
figure shows the form of curve representing equation (1) in rectangular 
coordinates. The even power oin-\-c involved in (1) causes the graph to 
be symmetrical with respect to the fine A V whose equation is ^ = —c. 
AV = \o. The branches of the curve are asymptotic to the line TT^ 
whose constant ordinate, Xo — /c Sr-i/tr= \t, equals the wave-length of 
the tail of the series. The points of inflection express the hypothesis 
that the intervals between consecutive lines of the same series attain 
a maximum, as n increases arithmetically, and then decrease indefi- 
