The Collected Mathematical Papers of Arthur Cayley volume 13

Cover The Collected Mathematical Papers of Arthur Cayley volume 13
The Collected Mathematical Papers of Arthur Cayley volume 13
Arthur Cayley
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M (r 1) n . Rv> r - r ~ l divisible by 1. 2 .... R; for then, writing NX for x, the form of the coefficient would have been and there would be no value (however great) of N by which the denominator factor w jr. R+i could be got rid of.
358 T941 941.
NOTE ON THE PARTIAL DIFFERENTIAL EQUATION [From the Quarterly Journal of Pure and Applied Mathematics, vol. Xxvi. (1893), pp. 1-5. ] IT is well known that this equation, R, S, T, U, V being any functions whatever of (x, y, z, p, q), in the case wher
...e u admits of an integral of the form uf(v) (u, v functions of x, y, z, p, q, and f an arbitrary functional symbol) can be integrated as follows; viz. Taking 7n l, m 2 as the roots of the quadratic equation (that is, writing m l + ra 2 = S and m 1 m 2 = RT UV), then, m^ denoting either root at pleasure, and ra 2 the other root of the quadratic equation, if the system of ordinary differential equations m^dx R dy + Udq = 0, - T dx + m^dy + U dp = 0, V dx + m 2 dq + R dp = 0, - Vdy + T dq + m-^dp = 0, p dx q dy + dz = 0, (equivalent to three independent equations) admits of two integrals u = const, and v = const.

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