The present paper is the second of a series of investigations into the anatomy of the lower Dicotyledons (1), aiming at the accumulation of anatomical data which, it is hoped, may prove of service in elucidating the phylogeny of the complex of orders usually regarded as situated somewhere near the base of the Angiosperm phylum. The continuance of the War has again materially retarded these investigations, and has also rendered it almost impossible to obtain material from abroad that would, doubtless, have been available in peace time. We are greatly indebted for substantial assistance in this relation more especially to Sir David Prain, F.R.S., Director of the Royal Botanic Gardens, Kew, and to Sir Frederick Moore, F.R.S., Director of the Glasnevin Gardens, Dublin, and to them we tender our grateful thanks.

The present paper is the third of a series of researches on the anatomy of the lower Dicotyledons undertaken with the object of seeing what, if any, aid anatomy might give in determining the point of origin of the so-called Monocotyledons. It deals more especially with the anatomical and histological peculiarities of the stem of the Calycanthaceæ.

Some months ago it was suggested that I should publish my thesis in its unfinished state. I refrained from doing so at that time, as I was unwilling to allow my work to appear until it was ready for publication. When it became apparent that the war. might continue indefinitely. I decided to publish the thesis in such a form as the exigencies of the case would allow.

That the stelar problem considered in this paper may be more clearly visualised, it will be well to divest of purely theoretical considerations the facts of stelar structure already known for Platyzoma, and to summarise such structural features as bear on the subject in hand.

In the first account of Platyzoma given by Robert Brown in 1810 (l), the habit of the stem and heterophyllic leaves, and the general form and position of the sporangia, were alone described. But in 1832 (2) a rough analysis of the structure of the stem was added, and the tubular nature of its stele was recognised. No material contribution to the knowledge of the stelar structure was made until 1893 (3), when Dr Poirault described the histology of the stele as shown by a small fragment of stem. In this material there was a sclerotic pith completely surrounded by an endodermis, outside which lay a zone of parenchyma surrounded by a broad ring of parenchymatous xylem. At the periphery of the xylem there was a narrow zone of phloëm surrounded by a large-celled pericycle and an outer endodermis. Neither leaf-gaps nor perforations were found in the stele, and accordingly the complete isolation of the pith from the cortex was recognised.

The intrinsic symbol of an amphicheiral knot is based on the idea of the sequence of the crossings; it replaces each letter of the alphabetical symbol by a number equal to one-half of the number of crossings intervening before the next occurrence of that letter as the knot is traversed in a definite direction. Hence it is seen that two knots, which have the same intrinsic symbol, are identical. Since the same number of pairs of crossings must elapse before the next occurrence of corresponding crossings of two identical knots when the knots are traversed in a given direction, it is seen that the converse is true also. It may be necessary to consider the complementary intrinsic symbol in order to detect identical knots.

In Part I a general account was given of the silicified peat-bed found at Rhynie, and one vascular plant was described in detail under the name of Rhynia Gwynne-Vaughani. Further study has shown that there are two species of Rhynia which we now distinguish as Rhynia Gwynne-Vaughani and Rhynia major. The account in Part I applies to both these species. Along with them there occurs a plant of similar grade of organisation to Rhynia, though quite distinct from that genus: this we name Hornea Lignieri. Asteroxylon Machiei, on the other hand, which will be described in the next part, was a plant of larger size and much more complex morphology.

We are thus now able to establish the existence and main features of four archaic Vascular Cryptogams from the Rhynie bed. In the present paper additional notes on R. Gwynne-Vaughani are given, and R. major and Hornea Lignieri described in detail. In conclusion, the morphological bearings of the facts will be briefly considered.

The theoretical determination of the longitudinal seiches of a lake was reduced by Professor Chrystal to the finding of those solutions of the differential equation

which vanish at x = 0 and x = a.

Here x denotes the area of the surface of the lake from one end up to any transverse section, and ranges from 0 to a, a being the total area of the surface of the lake; p(x) denotes the product of the area of the transverse section at x and its breadth at the free surface; λ=σ2/g, σ being the “speed” of the periodic motion and g the acceleration due to gravity; V denotes the total volume of water which has passed the section up to the time t.

Asteroxylon Mackiei was a plant of more complicated organisation and larger size than either Rhynia or Hornea, which have been described from the silicified peat-bed at Rhynie in the two preceding papers of this series. The generic name refers to the stellate outline of the xylem of the stem as seen in cross section, while the specific name commemorates the original discovery of the plant remains by Dr Mackie.