tained; the latter being the curve required, the former the property of the curve indicated by the given equation. 2. Again, if it be proposed to integrate the differential equation (x+ydy+zdz)=m* { (ydy−zdy)+(zdữ−xd^)+(xdy_yda)}, and at the same time to determine its geometrical interpretations, both immediate and remote, the same method can be employed with advantage. The equation obviously represents a curve of double curvature; and, remembering the expressions for the projections on the coordinate planes of the infinitely small triangle standing on the element of a curve with its vertex at the origin, we have at once for the immediate geometrical interpretation of the given equation Q=mP, Q and P having the same meanings as before. For the remote geometrical interpretation, or the integration of the given equation, we have whence rdr=mr2dp, r=Cem3, where is the whole angle swept out by the radius vector. 3. An inquiry obviously suggested by the last example, is the integration of the partial differential equation resembling it in form, namely, du du du): =m2 { x・ da dy du du dy and the determination of its geometrical interpretation, if it have any. The equation obviously represents some surface; and if both sides of the equation be divided by it will be easily seen that, being the angle between the radius vector to the point (x, y, z) and the perpendicular on the tangent plane at that point, the given equation is equivalent to r2 sin2=Q2=m2P2, or the given equation represents the surface for which the perpendicular from the origin on any tangent plane is in a constant ratio to the intercept between the point of contact and the foot of the perpendicular. Now putting and substituting, we get r2 = (1+m2)P2; r2 sin 0 and, remembering that in polar coordinates we get P= (2)* (dr)2 '+r2sin20+ sin20 (dir.) * = (1+m2),2 sin2 0, This, then, is the result of the transformation of the given equation to polar coordinates, the deduction of which by direct substitution would have been tedious and laborious, and the expression thus reduced may be integrated by series. It can be easily proved that the quadrature of the surface represented by this equation is given by the simple formula Σ=(1+m2) * SS r2 sin ✪ d0 do, a result which indeed might have been anticipated from geometrical considerations. 4. There are some differential equations whose integration may be facilitated by a partial employment of the above method. Thus if it be proposed to integrate the equation ydx—xdy= (x2 + y2)*dx, It is not difficult to reduce this solution to the form tan-1(2)=2 tan-(-1). 5. Again, let it be proposed to integrate the equation y(ydx—xdy) = (x2 + y2)e ̄%dx ; Thus the variables are separated, and the equation is reduced to an integrable form. 6. If the equation to be integrated were where, as before, the variables are separated, and the equation reduced to an integrable form. 7. There are some cases in which the solution of partial differential equations may be facilitated by transformation to polar coordinates. In these cases the partial differential equations reduce themselves to complete differential equations in a single variable. Thus the equation which is easily integrated, and ✪ replaced by x and y. 8. Let it be proposed to integrate the system of simultaneous partial differential equations Multiply the first equation by x, the second by y, and the third by z; add; and we obtain whence at once u=(a1yz+b1zx+c1xy) +uo where u denotes an arbitrary homogeneous function in x, y, s of the order zero. It is to be observed, however, that u cannot contain any inverse For instance, u cannot be of the form powers of x, y, z. ax+By yx + dy' as in that case u would not satisfy the separate partial differential equations. Thus the ultimate form of the required solution is u={(a1yz+b1zx+c1xy)+k, where k is an arbitrary constant. 9. Again, let it be proposed to integrate the system of partial differential equations d2w dx2 d2w where λ, μ, v are given constants; or, more generally, the system dy2 =cy2+ bx2+v. Multiply the first equation by x2, the second by 2xy, and the third by y2; add; and putting the symbol we obtain d d d Ꮖ +y +2 ▼(▼−1).w=(ax2+4bx2y2+cy1) + (λx2+2μxy+vy2), whence at once w='2 (ax +4bx2y2 + cy1) + }(λx2 + 2μxy+vy2)+u, +% where u1, uo are arbitrary homogeneous functions in x, y, z of the degrees unity and zero respectively. For the same reason as in the last article, u, and u。 cannot contain any inverse powers of x, y, z. Thus the ultimate form of the required solution is w=1'1⁄2 (ax1+4bx2y2+cy1) + 1⁄2 (λx2+2μxy+vy2)+(ax+By+yz)+8, where a, B, y, & are arbitrary constants. 10. Again, let it be proposed to integrate the system Multiplying the first equation by x2, the second by y3, and adding, as before, we get whence ▼(V+1).u=2(ax3+byx2+cxy2+dy3), u=}(ax3+byx2+cxy2+dy3)+a, the arbitrary constants in the second arbitrary function being of necessity reduced to cipher. Trinity College, Dublin, LIX. Experimental Relations of Gold (and other Metals) to Diffused particles of gold-production-proportionate size-colour -aggregation and other changes. AG GENTS competent to reduce gold from its solution are very numerous, and may be applied in many different ways, leaving it either in films, or in an excessively subdivided condition. Phosphorus is a very favourable agent when the latter object is in view. If a piece of this substance be placed under the surface of a moderately strong solution of chloride of gold, the reduced metal adheres to the phosphorus as a granular crystalline crust. If the solution be weak and the phosphorus clean, part of the gold is reduced in exceedingly fine particles, which, becoming diffused, produce a beautiful ruby fluid. This ruby fluid is well obtained by pouring a weak solution of gold over the phosphorus which has been employed to produce films, and allowing it to stand for twenty-four or forty-eight hours; but in that case all floating particles of phosphorus should be removed. If a stronger solution of gold be employed, |