MATERIALS FOR A MONOGRAPH OF THE ASCONS. 341 
indeed they ever occur.!_ Here we must consider the remark- 
able facts discovered by von Ebner concerning the relations 
of the triradiate systems to crystalline symmetry. 
Von Ebner divides the triradiate systems of calcareous 
sponges into three main classes, which he terms perregular, 
sagittal, and irregular respectively. In the perregular 
triradiates, exemplified by those of Clathrinide, the optic 
axis is vertical to the facial plane of the rays, and all three 
rays are optically and morphologically equivalent? and meet 
at equal angles. In the sagittal spicules, seen in Leucoso- 
leniidez and Heteroceela, the optic axis 1s never vertical 
to the facial plane, and there are always an unpaired and 
two paired rays; the morphological axis of the unpaired ray 
lies in a plane of crystalline symmetry which contains the 
optic axis and halves the angle between the two paired rays. 
An irregular triradiate is defined as one in which no ray can 
be found, the morphological axis of which defines a plane of 
crystalline symmetry containing the optic axis and halving 
the angle between the other two rays; but since no irregular 
spicules, in this sense, could be discovered, it is doubtful if 
they exist at all, and it is not necessary to consider them 
further. The sagittal triradiates were further divided by 
von Ebner into primary and secondary forms. In the 
primary sagittal triradiates the morphological axes of all 
three rays lie entirely in planes of crystalline symmetry 
1 As von Ebner has pointed out, the three rays of a triradiate system 
seldom lie in one plane, but form the edges of a shallow pyramid, of which the 
base lies towards the gastral, the apex towards the dermal, surface. Hence a 
careful distinction must be drawn between the actual angles at which the rays 
join, and the apparent angles which they present in the facial projection, that 
is to say, when a spicule is seen under the microscope with its dermal surface 
uppermost and the apices of its rays resting on the slide. In the case of 
certain triradiates of Leuconia solida, von Ebner has shown that a spicule 
which appears regularly equiangular when viewed in the facial projection, may 
be, in reality, markedly sagittal if its real angles be measured accurately. 
2 One ray, as in Clathrina blanca and C. lacunosa, may be longer 
than the other two, without disturbance of other conditions. Von Ebner does 
not appear to have examined triradiates of this type. 
