1. Since I had the honour of showing the phonograph to the Royal Society of Edinburgh, at a special meeting in November 1894, the instrument has occupied a good deal of my time and attention, and I now venture to give the general results of the investigation.

2. The instrument chiefly studied has been the machine used in this country known as the “Commercial Phonograph.” Any records taken by myself have been obtained with the ordinary apparatus forming part of the “commercial” speaker arm, but I have always reproduced these with the aid of the so-called “musical” arm. The commercial machine, or, to give it a better name, the English model, is so geared that the wax cylinder, inch (197 mm.) in circumference, makes two revolutions in one second, while the spiral grooves described on the cylinder are inch (⅛ mm.) apart. A spiral line about 136 yards in length may be described on the cylinder, and the recording or reproducing point travels over this distance in about six minutes.

3. I have also used the American model, which resembles in all essential particulars the one just described, except that the grooves on the cylinder are inch (¼ mm.) instead of .

The genus Anaspides was founded in 1894 by Mr G. M. Thomson of Dunedin, New Zealand, for the reception of a very remarkable Schizopod Crustacean which he had discovered in a fresh-water pool at an altitude of 4000 feet on Mount Wellington in Tasmania. The very striking peculiarities of the animal, the absence of a carapace, the presence of plate-like gills attached to the bases of the thoracic legs, and the possession of an auditory organ in the peduncle of the antennules, led its discoverer to regard it as the type of a new family of Schizopods, the Anaspidæ, while suggesting that it might be entitled to “even higher specific rank.” I have had an opportunity of examining three specimens of Anaspides presented to the Museum of University College, Dundee, by Dr Chas. Chilton of New Zealand, and from the dissection of one of these I have been able to supplement, in some important points, Mr Thomson's account of the external anatomy of the animal. I wish also to call attention to the remarkable resemblance, indicative I believe of close affinity, which Anaspides bears to certain Palaeozoic Crustacea belonging to a group hitherto supposed to be unrepresented among living forms. The present paper has been prepared under the direction of Prof. D'Arcy W. Thompson.

The theory of the singular solutions of differential equations of the first order, even in the interesting and suggestive form due to Professor Cayley (Mess. Math., ii., 1872), as given in English text-books, is defective, inasmuch as it gives no indication as to what are normal and what are abnormal phenomena. Moreover, Cayley added an appendix to his theory regarding the circumstances under which a singular solution exists, which is misleading so far as the theory of differential equations is concerned, if not altogether erroneous.

The main purpose of the following notes is to throw light on the point last mentioned by means of a number of examples. I have also taken the opportunity to furnish simple demonstrations of several well-known theorems regarding the p-discriminant which do not find a place in the current English text-books.