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angles applied arbitrary constants assume asymptotes asymptotic curves axes becomes centre of gravity change sign co-ordinates coefficients condition conic section Consequently considered corresponding curve deduced definite integrals degree denote determine differential equation eccentric anomalies elimination ellipsoid equal exact differential expression finite force formulae given Hence homogeneous function infinite intersect isothermal limits linear lines of curvature magic square magnet Mathematical Journal method molecules motion Multiply negative obtained orthogonal paper parallel partial differential partial differential equations perpendicular plane positive problem quantities respectively result roots rotation satisfied second order shew ſº substituting suppose surfaces of revolution symbol symmetrical system of equations tangent theorem theory tion transformation Trinity College values vanish variables velocity whence zero
Page 63 - dx dx dy dy dz dz Differentiating the first of these equations with respect to x, the second with respect to y, and the third with respect to z ; putting x, y, and z each = 0, in the result, and making use of equations
Page 69 - A/"? T , &c. will be ultimately divergent. Hence, in this case, for any time, however small, before the initial instant, the distribution will be impossible, and therefore the given distribution cannot be any stage but the first in a system of varying temperatures. 2. If the convergence of the series for 0
Page 64 - Now 0 may be any point in one of the surfaces (a), and therefore each of these surfaces has its lines of curvature traced upon it by the surfaces of the series (a^), (a 2 ). Similarly it may be shewn that each surface, (aj has its lines of curvature traced by
Page 280 - that the projection of any line of curvature on the plane of the greatest and mean, or of the mean and least axes, does not meet its consecutive. It may be remarked with respect to equations (4), that any one of them may be deduced from any other, by combining it with the equation u
Page 281 - that the lines of curvature of any surface of the second order with a centre, are its intersections with confocal surfaces of the second order. Since this property is independent of the centre, it follows that it must also be true for the case of a surface without a centre.
Page 227 - Hence any two of the axes determined by equations (6) and (5) are at right angles, and therefore the three must form a rectangular system. If we take it for axes of co-ordinates, and if P, Q, B be the three roots of (6), the equation to the surface becomes
Page 145 - It may be well to say a few words of the history of this part of mathematics. One of the first results of the differential notation of Leibnitz, was the recognition of the analogy of differentials and powers. For instance, it was readily perceived that
Page 38 - in such a manner that (7) shall be satisfied ; in fact a series of parallel planes is necessarily isothermal. We have thus seen that the two conjugate orthogonal series to a series of isothermal cylinders are themselves isothermal, which agrees with what was proved in the paper already alluded to, (vol. in. p. 286).
Page 228 - be all positive; a hyperboloid of one sheet if one of them only be negative, and a hyperboloid of two sheets if two of them be negative. If all three be negative, the surface will be imaginary. Hence, from equation (6) we infer that if /, g,
Page 225 - shall disappear in the transformed expression H^. This will be ensured if the product x'y vanishes for every point in the plane x'y\ and for every position of this plane passing through the principal axis OX' | a definition equivalent to the one in which a principal axis is defined as an axis which is perpendicular to its diametral plane.