The present Memoir contains an investigation of the differential equation (of the second order) of the geodesic lines on a surface, the coordinates of a point on the surface being regarded as given functions of two parameters p, q, and researches in connection therewith; a deduction of Jacobi's differential equation of the first order in the case of a quadric surface, the parameters p, q being those which determine the two sets of curves of curvature; formulas where the parameters are those

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... h determine the two right lines through the surface; and a discussion of the forms of the geodesic lines in the two cases of an ellipsoid and a skew hyperboloid respectively. Preliminary Formtdce. 1. I call to mind the fundamental formulas in the Memoir by Gauss, Disquisitiones generates circa superficies curvas, Comm. Gott. recent, t. vi., 1827, (reprinted as an Appendix in Liouville's edition of Monge,) together with some that I have added to them. The coordinates x, y t z of a point on a surface are regarded as given functions of two parameters P, q, these expressions of x, y, z in effect determining the equation of the surface, and we have dx+\d?x = adp + a'dq + £ (adp 2 + 2a dpdq + a'dq 2 ), dy+\d 2 y = 6 dp + b'dq+1 (/3 dp 1 + 2/3'dp dq+p'dq*), dz +\d 2 z = cdp +cdq + £ (ydp % + 2ydpdq + y"dq 2 ), A, B, C = I'c '-b'c, ca'-ca, ab'-a'b' } whence differential equation of surface is Adx+Bdy + Cdz = 0. Also E, F, G = a 3 +b 2 + c\ aa + 66'+cc\ a'* + 6' a + c 2 ; so that element of length on the surface is given by dx 2 + dy 2 + dz* = "Rap* + 2¥dp dq + Gdq 2 ; or as I write it, = (E, F, G^dp, dq) 2 ; and moreover Y 2 = A a +B 2 +C 2 = E G -F 2 . The equation (E, F, GQdp, dq) 2 = 0 determines at each point on the surface two directions (necessarily imaginary) which are called the " circular" directions. Passing on the surface from point to point along the circular directions, we obtain two series of curves (always imaginary) which are the "circular" curves; the equation (E,F,G][djj,d2) 8 = 0 is the differential equation of these curves; and if we have E = 0 , G=0, then this becomes dpdq=0; viz., we have in this case ^?=const. and q=const, as the equations of the two sets of circular curves re-