. Carnegie Institution of Washington publication. EXAMPLES OF SOLENOIDAL FIELDS. 35 The line-integral of the normal component of the vector is easily found for any closed curve having the form of a rectangle with sides parallel to the axes of coordinates. If two of the sides are the coordinate axes, and the two others the lines x = x andy = y, the line-integral taken over the closed curve will be Axy-\-AyX. Substituting the values (a) of the components, we get the line-integral equal to axy+bxy, and the solenoidal condition is seen to be fulfilled if b = —a. Thus the formulae will be (b) Ax =
. Carnegie Institution of Washington publication. EXAMPLES OF SOLENOIDAL FIELDS. 35 The line-integral of the normal component of the vector is easily found for any closed curve having the form of a rectangle with sides parallel to the axes of coordinates. If two of the sides are the coordinate axes, and the two others the lines x = x andy = y, the line-integral taken over the closed curve will be Axy-\-AyX. Substituting the values (a) of the components, we get the line-integral equal to axy+bxy, and the solenoidal condition is seen to be fulfilled if b = —a. Thus the formulae will be (b) Ax = ax Ay = —ay Fig. 36 represents the components of this solenoidal vector. If the vector represents velocity, the motion given by fig. 36 will be the simplest typical fluid motion pro- ducing a deformation of the fluid masses without change of volume. A vector-line is determined by the condition that the projections dx and dy of its line-element are proportional to the vector-components Ax and Ay. It is there- fore given by the differential equation (c) dx ^dy AX Ay Substituting the values of Ax and Ay according to (b), and integrating, we find (d) xy = const. i. e., the vector-lines are equilateral hyperbolae, having the axes of coordinates as asymptotes (fig. 37). These axes themselves belong to the system of lines of flow,. Fig. 37.—Hyperbolic lines of flow and circular curves of equal intensity I, 2, 3, . . of a plane Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Carnegie Institution of Washington. Washington, Carnegie Institution of Washington
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