556 DR J. HALM ON 
5. Cxesium—continued. 
Sussiprary Series (1st Component), 
log a,=4'90651 — 10 
log b, = 521884, — 10 al ca 
Limits of 
(m+p)|  v obs. v comp. one ~ | Error of 
ig Obs. 
4:0 10855°6 | 10855 °6 0:0 0:8 
5:0 14339°2 | 14343°1 -3°9 10:0 
6:0 16094°2 | 16094 3 — O01 1:3 
7:0 17108°4 | 17106-0 +2°4 15 
8:0 17745°9 | 177456 +0°3 16 
9-0 18175°5 | 18176°7 -—12 q 
10:0 18481:°2 | 18481°4 - 0:2 q 
11:0 18705°9 | 18705-0 +09 q 
I have omitted the second subsidiary series of Rubidium, of which only three lines 
are known, the series being thus insufhcient for determining all the constants. It 1s, 
I think, obvious, and scarcely requires to be mentioned, that the principal series of 
Rubidium and Cesium, in which the four unknowns of our equation had to be 
computed from the only four lines available, can tell us nothing of the accuracy of the 
formula employed. But for obvious reasons I have made it a rule to compute every 
series from which all the four constants may be obtained. 
Let us now, before we proceed to other groups of elements, investigate the residuals 
given in the columns [Obs. — Comp. ], by comparing them with those of the hitherto best 
empirical formula, that proposed by KaysEr and RUNGE: 
108\-l=a4+6m-2+em-4. 
This equation apparently contains three unknown constants, a, b, and ¢, and seems 
therefore to possess in this respect an advantage over ours, which has four unknown 
quantities. But it is well to consider that a fourth unknown is implicitly involved in 
Kayser’s formula, viz. the value of m for the first lime of the series, which we may 
call m,. Strictly speaking, the difference between Kayser’s formula and the one here 
proposed is therefore this, that the fourth unknown m, in the former is an integer, 
while in the latter it is a compound fraction. But let us grant to Professor Kaysmr’s 
equation the full advantage of having one unknown less than ours. If it would 
represent the observations equally well, it would doubtless be the superior formula. 
This condition, however, is far from being fulfilled. Let us, for instance, consider the 
principal series of Potassium. Professor Kayser has computed the three unknowns 
of his equation by the method of least squares, and has found the residuals, expressed 
in units of wave-lengths, which are given in the first column of the following table, 
