By Watson E.J.

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Isolated * Ueber die Vieltheiligkeit der ebenen algebraischen Cttrven, Math. Annalen, Vol. io (1876), pp. 189-198. t Ueber die reellen Züge algebraischer Ctirven, Math. Annalen, Vol. 38 (1891), pp. I15-138. % Eine neîte Relation zwischen den Singularitäten einer algebraischen Cttrve, Math. Annalen, Vol. 10 (1876), pp. 199-209. ALGEBRAIC CURVES AND SURFACES. 29 real double tangents, besides imaginary double tangents ; also w] real inflexions, besides imaginary inflexions. Then it can be proved by means of the principle of continuity, that the following relation must hold : n + 7£j'+21" z=le + rf + 2 d".

The roots of an algebraic equation bßm + bß™-1 + b2ßm~2 + ... + bm = o, h bp bv ... bm being integers. * 1893, No. 2, p. 113. , No. 4. J Comptes rendus, 1893, p. 1040. § Vol. 43 (1894), pp. 216-224. 54 LECTURE VII. It will be noticed that the latter proposition really includes the former as a special case ; for it is of course possible that the /3's are rational integral numbers, and whenever some of the roots of the equation for ß are equal, the corresponding terms in the equation (2) will combine into a single term of the form akeßk.

Annalen, Vol. 42, pp. 377-408. 40 LECTURE V. This treatment of the functions defined by linear differential equations of the second order is of course only an example of the general discussion of complex functions by means of geometry. I hope that many more interesting results will be obtained in the future by such geometrical methods. V I . : ON T H E M A T H E M A T I C A L C H A R ACTER OF SPACE-INTUITION AND THE RELATION OF PURE MATHEMATICS TO T H E APPLIED SCIENCES. ) IN the preceding lectures I have laid so much stress on geometrical methods that the inquiry naturally presents itself as to the real nature and limitations of geometrical intuition.