The relative motion in a group of three such sector-shaped cylinders rotating about their 

 common edge is shown in Figure 171. 



Figure 171 —Streamlines for the motion of 

 fluid relative to the walls within a 

 threefold sector-form cylinder 

 rotating about its apex. 

 See Section 102. 

 (Copied from 

 Reference 10.) 



If a = n/i or 37t/4, cos 2a =0; but then, also, either the first or second A. 



be- 



comes infinite. To handle such cases, a modified formula must be developed by seeking the 

 limit forms that <fe and ip take on as a approaches the value stated. 



In Equation [I7d] for the kinetic energy, or T^ = {t/2)p fcp q^ds, along the side at 

 ^ = -«)?„ = <^'") (is - dr, and, since sin [(2n + 1)77/2] = (-1)", 



</> = 0_« =- c^ '•' tan 2a + a, a2 s (-1)" A^^ ^ , (^) 



(2n + l) 



On the other side, where 6 = +a , ^ and q are both reversed in sign, hence the integral has 

 the same value as on the first side. Finally, over the curved end q = 0. Hence the kinetic 

 energy per unit length is f^ = p oj /° ^_„^ dr, or, after integrating, 



T, =pco^ a^\ — tan 2o< + 2 (-1)" 



(2n + 1) — +2 



[102d] 



A semicircle is obtained if a = n-/2; it is revolving about the central line of its base, 

 as in Figure 170b. In this case, since (-1)^" = 1, using Equation [102b], 



252 



