Algebra und Zahlentheorie A MATRIX REPRESENTATION OF ASCENDING AND DESCENDING CONTINUED FRACTIONS By L. M. MILNE-THOMSON, Greenwich The elements of a matrix, which is equal to the continued product of suitably chosen matrices, are shown to have the properties of the convergents of a continued fraction. The method leads at once to a generalised definition of continued fractions in many dimensions. By turning through two right angles the plane on which an ordinary (two dimensional) continued fraction is written, an ascending continued fraction composed of the same elements is obtained. This again can be represented by a product of matrices and generalised to many dimensions. Some properties and applications of continued fractions of both types are considered. A paper dealing with this subject will be published in the Proceedings of the Edinburgh Mathematical Society. LE SERIE RICORRENTI ASSOCIATE DEL 2» ORDINE (Generalizzazione delle Un e Vn di Lucas) Di GIACOMO CANDIDO, Brindisi i- Defin. Due serie del 2° ordine WB, wlt entrambe riferite ...., WH, ....(W) ¥0 , Wlt ...., Wn , ....(¥) alla stessa equazione g—PÌ^ caratteristica Ç = o .... [ i ] , coi rispettivi valori iniziali W0, Wx ; ¥0, Wx le diremo associate se una coppia qualunque ( Wz-, ¥z- ) di termini corrispondenti verifica la identità Vnp,q)-àWi(p,q) 2 con A =p = kq<, ....[E] — 4 q9 e k costante. E s : i) Le Vn ed Un di Lucas. 2) Le soluzioni della equazione x2 — Ay1 = kq* 2. L'integrale generale della [W) è dato da (Eulero, Le Longchamps, d'Ocagne): Wn (p, ç)=W±l7n — p W0 £/„_! 27