mm libeaey S. PATENT OFFICE, Class V O' o o. V V v o • j vj ^ sec c< c 3c< c=K’> (i) then, (1) divided by X2 becomes _1_ dy_ X2’ da* X3 ’ dx K2y = 0 (2) Now, if L = ±K, M=:qFK, then (t4+l)(t4 + m>=0 <3> is the symbolical decomposition of (1) or (£). When L and M are any constants whatever, (3), when developed, gives rise to a linear differential equation of the second order reducible to an equation with constant coefficients by changing the independent variable from x to t , where 39 In deducing from (3) its development, the order of the symbolical factors is indifferent, but the two particular inte- grals of the development are, I think, obtainable by reversing in (3) the order of the symbolical factors. The differential resolvent of every such trinomial cubic as that discussed by Mr. Hawson is soluble by a change of the independent vari- able, and belongs moreover to a comparatively simple form of equation soluble by such change. The theory of Transcendental Solution has led me to the following proposition (theorem) : — If an irreducible algebraical equation of the degree n have a homogeneous linear differential coresolvent of the order m, then any root whatever of the algebraical equation can be expressed as a linear and homogeneous function of any other m of its root . The general demonstration would not be much more diffi- cult than or very different from the particular demonstration of the case m- 2. The converse of this theorem, I believe, is true. In such case let a and b be the particular integrals of the differential resolvent which (since m~ 2) is by hypothesis„of the second order only. Let a, |3, and y be any three of the roots of the algebraical equation. Then, since among the values that can be assigned, by means of the arbitrary constants, to the general integral, the roots of the algebraical equation are included, we have three such relations as Aa + D5 = a* B a + E 6-/3, Ca + F6 = y, 40 wherein A, B, ,F are constants. Multiplying the first, second, and last of these equations into the arbitrary multi- pliers X, ju, and v, and adding the results, we have (XA + /jlB + vC)a + (XD + /xE + vE)6 — Xa + gf 3 + vy (4) Hence if the ratios of any two of the quantities X, /x, v to the third be so assigned as to satisfy the equations \A + fiB + vC - 0, \J) + gE + v¥ = 0, then the sinister of (4) will vanish independently of a and 1, and the homogeneous linear relation Xa 4- yu/3 4- vy -• 0 (5) will subsist among the roots a, j3, and y of the algebraical equation. When n~ 3 we (since the differential resolvent is homogeneous) have without reference to, but consistently with, the theorem, a + /3 + y — 0 (6) Combining the above theorem with one given by Abel and Galois, we conclude that : — If qn algebraical equation ham a differential resolvent of the second order , the algebraical equation is resoluble algebraically . Before closing I would add that, as it seems to me, it would be more consonant with the notation and practice of the rule of three, and, therefore, with convenience and the analogies of arithmetic, if by the ratio p : q there were univer- sally understood (not the fraction p-^q, but) the fraction q-p. 41 Mr. R. D. D^rbishiee, F.G.S., read a paper entitled “ Notes on Marine Shells found in Stratified Drift at Maccles- field,” and exhibited a series of specimens. The specimens were chiefly collected by Mr. W. J. Sainter and Mr. Lowe, of Macclesfield, from sand and gravel exposed in the formation of the new Cemetery on the north side of the town, at an elevation of between 500 and 600 feet above the level of the sea. Unfortunately the buying of specimens had caused the intrusion of many spurious fragments, casting suspicion on several that might after all prove to be genuine. The beds in question were exposed on a south-easterly face, but are now defaced by ballast tips ; consist of fine (running) sand, fine and coarse shingle, and very coarse gravel with large pebbles unscratched ; and, while stratified, in general horizontally, exhibit in their great irregularities of extension, level and false bedding, characteristically marine aspect, as of a sea bottom under the influence of tidal and other varying currents. Below appears the “lower boulder clay” of the Ordnance geologists. The shells are nowhere numerous. Mr. Lowe speaks of finding some in layers. Unfortunately the shells from parti- cular beds have not been distinguished. In the list specimens obviously spurious have not been noticed. The following species had been identified : — 42 Pholas crispata. Pholas Candida. Mya truncata. Mya arenaria. Psammobia ferroensis. Donax anatinus. Tellina solidula. Mactra solida. Lutraria elliptica. Cytherea chione. Venus striatula. Artemis lincta. Cyprina islandica. Astarte elliptica. Astarte arctica. Cardium echinatum. Cardium aculeatum (?) Cardium rusticum. Cardium edule. Cardium Norvegicum. Mytilus edulis. Modiola modiolus. Nucula sp. Area lactea. Pectunculus sp. Pecten opercularis. Ostrea edulis. Patella vulgata. Dentalium entale. Dentalium Tarentinum. Trochus cinerarius. Littorina littorea. Littorina rudis. Littorina littoralis. Turritella communis. Aporrhais pes pelicani. Natica nitida. Natica monilifera. Murex erinaceus. Purpura lapillus. Nassa reticulata. Nassa incrassata. Buccinum undatum. Fusus gracilis. Fusus antiquus. Trophon clatliratum. Mangelia turricula. Mangelia rufa. Mangelia nebula. Cyprsea Europsea. Total, 50 species. Mr. Darbishire compared this list with those of the Moel Tryfaen drift (Caernarvon), and the Kelsey Hill (Hull) fossils, and with several lists of recent faunas of the British 43 Eastern and Western seas, and of seas North and South of the British Isles. The present list was specially remarkable for including Cytherea chione, Cardium rusticum, Cardium aculeatum (?), and Area lactea ; all of them shells reaching their highest northern range in the extreme south or west of England and Ireland, a circumstance believed to be new to the history of the so-called