as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term m[zeta] in w. Similarly, with x + yi = c[root][sin ([xi] + [eta]i)] (17) the function [psi] = Qc sh ½([eta] - [alpha]) sin ½([xi] - [beta]) (18) will give motion streaming past the fixed cylinder [eta] = [alpha], and dividing along [xi] = [beta]; and then x² - y² = c² sin [xi] ch [eta], 2xy = c² cos [xi] sh [eta]. (19) In particular, with sh [alpha] = 1, the cross-section of [eta] = [alpha] is x^4 + 6x²y² + y^4 = 2c^4, or x^4 + y^4 = c^4 (20) when the axes are turned through 45°. 33. _Example 3._--Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function [psi]1 is to be made to satisfy the conditions (i.) [nabla]²[psi]1 = 0, (ii.) [psi]1 + ½Rx² = ½Ra², or [psi]1 = 0 when x = ±a, (iii.) [psi]1 + ½Rx² = ½Ra², [psi]1 = ½R(a² - x²), when y = ± b. Expanded in a Fourier series, 32 __ cos (2n + 1) ½[pi]x/a a² - x² = ----- a² \ ---------------------, (1) [pi]³ /__ (2n + 1)³ so that 16 __ cos (2n + 1) ½[pi]x/a · ch(2n + 1) ½[pi]y/a) [psi]1 = ----- a² \ ---------------------------------------------, [pi]³ /__ (2n + 1)^3 · ch(2n + 1) ½[pi]b/a 16 __ cos (2n + 1) ½[pi]z/a w1 = [phi]1 + [psi]1i = iR ----- \ ------------------------------, (2) [pi]³ /__ (2n + 1)^3 ch(2n + 1) ½[pi]b/a an elliptic-function Fourier series; with a similar expression for [psi]2 with x and y, a and b interchanged; and thence [psi] = [psi]1 + [psi]2. _Example 4._--Parabolic cylinder, axial advance, and liquid streaming past. The polar equation of the cross-section being r^½ cos ½[theta] = a^½, or r + x = 2a, (3) the conditions are satisfied by [psi]´ = Ur sin [theta] - 2Ua^½ r^½ sin ½[theta] = 2Ur^½ sin ½[theta](r^½ cos ½[theta] - a^½), (4) [psi] = 2Ua^½ r^½ sin ½[theta] = -U[root][2a(r-x)], (5) w = -2Ua^½ z^½, (6) and the resistance of the liquid is 2[pi][rho]aV²/2g. A relative stream line, along which [psi]´ = Uc, is the quartic curve (4a²y² - (y - c)^4 4a²y² + (y-c)^4 y - c = [root][2a(r - x)], x = -------------------, r = ---------------, (7) (4a(y - c)² 4a(y - c)² and in the absolute space curve given by [psi], dy (y - c)² 2ac -- = - --------, x = - ----- 2a log (y - c). (8) dx 2ay y - c