The object of this paper is twofold: in the first place, to classify and sytematise vector differential equations; and in the second place, to show the applicability of quaternions to the theory of differential equations. So far as I am aware, the only paper of importance on the subject is one by Tait. He, however, deals only with simple cases.

In the classification of forms I have followed in general the treatment adopted by Forsyth. Reference will also occasionally be made to Jordan. In what follows the order of the highest occurring differential coefficient will be called the rank of the equation; while the order of the equation will be used, as usual, to denote the order of the equivalent normal system.

Gemünden is situated in Rhenish Prussia, about eighteen miles to the west and slightly also to the south of Bingen, in the district known as the “Hunsrück,” and the rock in which the fishes to be described in this memoir occur is called the “Hunsrück Slates.”

These Hunsrück slates belong to the Lower Devonian of the Rhenish area, and the position assigned to them by German geologists is as follows:—

The above table is quoted from Kayser and Lake's Text-book of Comparative Geology, as is also the following brief statement regarding the Hunsrück slates themselves.

This communication deals with the fossil plants of the Calciferous Sandstone Series and Carboniferous Limestone Series of Eskdale and Liddesdale; of the Lower Coal Pleasures of Canonbie; of the Middle Coal Measures of Byre Burn, Canonbie; and of the Upper Coal Measures, Joekie's Syke, Cumberland; to which is added a list of the fossil plants of the Calciferous Sandstone Series of Cumberland and Northumberland, with the object of comparing them with those from the corresponding series in Scotland.

The Canonbie Coalfield occupies a small tract of ground between the Liddel Water and the river Esk in the south-east part of the county of Dumfries. Though of limited extent, the coalfield has aroused considerable interest, due partly to the important series of plants obtained from the beds, and partly to the questions bearing on the correlation of the Carboniferous rocks of the Scottish Border with those in the North of England and Central Scotland.

Since the publication of my “Report” in December 1899, Mr Tait, fossil collector to the Geological Survey of Scotland, has been successful in adding a number of specimens of great interest to the valuable series of Scottish Silurian Fishes which supplied the material for that memoir, and with the permission of the Director of the Survey, to whom my best thanks are due, I propose to give descriptions of them in the following Supplement. The search for these remarkable fishes has likewise been taken up by private collectors, and I have therefore pleasure in acknowledging my indebtedness to Messrs J. Young, of Lesmahagow, P. Macnair, of Glasgow, A. Whyte and F. White, of Muirkirk, and D. Nimmo, of Hamilton, for kindly submitting to me the material resulting from their own work in this field.