and 



_ U b (a + b) cos a 

 u = + g^ = , L84j] 



Jy^ + c^ i\y\ + V/ + c2) 



, U a (a + b) sin a 

 ^ = i?„= , ^ . . [84k] 



Jy^ + c^i\y\+Jy^ + c^) 



The formulas for this case and the preceding are readily shown to differ by terms representing 

 a uniform flow, with use of the fact that (a + 6) (a - 6) = c^ = (c'+ b') (a'- 6'). 



If there is no circulation about the cylinder, F = 0. 



For motion parallel to the major axis, a = or n-; to the minor axis, a = n-/2 or 37t/2. 

 If F = 0, the geometrical flow net is the same for flow parallel to either axis; the c/i curves 

 for one case become the ip curves for the other, and ^, ip and all velocities are changed in a 

 uniform ratio. The general case, for which the formulas have been written, can be regarded 

 as formed by the superposition of these two simpler cases. 



Furthermore, if the motion is parallel to an axis, and if F = 0, the velocity at a given 

 external point is the same for all confocal forms of the cylinder. For the relations between 

 X, y, and ^, rj are unaffected so long as the foci are not disturbed; and changing a and b 

 merely multiplies 0, ip and hence all velocities by a uniform factor. 



Figure 138 will serve to illustrate the flow for motion parallel to either axis. The 

 ellipse drawn as a broken curve, or any other ellipse confocal with it, may represent the 

 cylinder. The foci are at the ends of the horizontal heavy line. Either family of curves, 

 that crossing the vertical or the horizontal axis, constitutes streamlines according as the 

 motion is parallel to the major or to the minor axis; the curves of the other set are then the 

 equi potentials. The arrows on the curves refer to motion along the minor axis. 



No similar identities occur in motion oblique to the axes. 



The kinetic energy of the fluid, per unit length of the cylinder is, by Equation [17d], 

 when F = 0, 



P [ P^ a+ * -7 A f 9 



Tj = (pdip = e ^0 (6 cosa cos r] + a sin a sin t]) drj 



•^ 



or, using Equation [83d, e], 



Tj = — pU^ (b^ cos^ a + a^ sm^a ). [841] 



The forces are as in the last case, Section 83. 



(For notation and method; see Section 34; Reference 1, Article 71; Reference 2, 

 Section 9.65; Ratib^°'*; Krienes^"^.) 



203 



