Dr. Schumann's Formulae for Diatom-lines. Bij W. J. Hichie. 9 
mark of the species, and determined the number of such strife there 
should he in a Paris line. 
For example, if there be 30 such transverse strife in rJV", then 
30 « = i^o'" ; therefore « = ^W". 
If we let a denote the number of these transverse strife in 
a certain length, taking either too of a Paris line, or XT jVo °f an 
Fnglish inch, or any other standard of measurement, and denote 
the standard itself by E, then 
E 1 a 
(( . « = E , a — — > — = rr • 
a U. E 
If we call the corresponding numbers of the vertical and 
oblique striae b and c respectively, then 
1 - 6 and I - £ 
h ~ E ’ dn 7 — E 
It follows now simply from Figure 1 that 
7 . A B ~ a. . ji , y = — ^ ’ 
\f a 2 + /S’" 
We see now that the line-number, c, is more directly dependent 
on the line-numbers, a and b, than 7 on a and /3. And herein lies 
the value of the above given definition of the line-number, inde- 
pendently of the fact that the magnitudes a, b and c are to be 
directly considered, whilst a, /3, 7 have to be deduced from them 
afterwards. 
We may remark also that the numbers of the lines stand in the 
same relation to one another as the corresponding sides of the 
triangle, inasmuch as both are inversely proportional to the heights 
of the triangles. 
Yet (setting aside the three primary series) others also appear, 
which I call secondary, namely, the series A D, BE, C F. 
Now, if we call the numbers of the lines corresponding to them 
d, e. f, we easily find 
d = \/ a 2 -j- 4 6 2 , c = \t i d 1 + 6 2 , / = c. 
In Figure 2 the direction of the above-mentioned series is 
denoted by the lines U A, U B, . . . . U F. 
To these, as the figure shows, two others are added, namely, 
U D' and U E'. 
