Br. Schumann's Formulas for Biatom-lines. By W. J. Hickie. 13 
I find the average values to be 
For Tahellaria fenestratata^ a = 33, 6 = 32, 
„ Gomphogramma rupestre^ a = 33, h = 29. 
The triangle ABC, which I take as the basis of the structure, 
is in this case isosceles.* 
The directions of the three primary systems of lines are deter- 
mined by the sides of the triangle, and the transverse lines by C B, 
and the inclined lines by 0 D (or A B) and C A. 
The three secondary systems of lines have the directions A D, 
B E and C F. The longitudinal lines D A, P Q, H F, &c., conse- 
quently appear here as secondary. 
For this structure the following formulae hold good : 
d = ^4 62 - e =f = /v/2^M^, b = c = i Va^ + d\ 
where a denotes the number of the transverse lines, and d the 
number of the longitudinal lines which appear in a given space,— 
say y^77 of a Paris line. 
In Case 1 we found the following rules : If we square the 
numbers of the transverse lines and of the longitudinal lines, and 
add these squares, and then extract the square root, we get the 
line-number for the primary oblique system. Here in Case 2 we 
must further divide the number so found by 2. This is the 
essential difference between the corresponding and alternating 
series and their structures. 
If we be in doubt to which class the species of diatom belongs, 
we may employ this means to decide the matter, when the single 
puncta are indistinct. 
For example, if, in any species, we have found 30 as the number 
of the transverse lines, and 40 as the number of the longitudinal 
lines, and lastly, 50 as the number of the most distinctly appearing 
oblique lines, then we are certain that the series of puncta are 
corresponding series. 
If, in another species, we find 30, 40, and 25 as the correspond- 
ing line-numbers, the structure consists of alternating series of 
puncta. 
If we take (j>u (l>2, • . • </)6 to denote the smallest angles which 
the six systems treated of form with the axis, then we get 
A a 
<t>i = 90°, = <l>3 = -2 » sin- <t>2 = sin- ^3 = 26' 
3a ^ A 
sm. (pa =■ sin. = — ■ , tan. cb. — tan. cb = S tan. -- ? 
2 V2 a2 + 62 2 
* The diagrams pertaining to this part have been unavoidably omitted for 
typographical reasons. 
