RECENT PROOBESS IN HYDEODYNAMICS. 45 



When the boundary consists of two intevsccting circles, the deter- 

 mination of the motion can bo made in a manner similar to that for two 

 non-intersecting circles. When the two circles ai'e equal and pass 

 through each other's centres, the velocity potential for the system moving 

 parallel to the line of centres is f = — aW (r~^ + r''^), where r. r' are 

 the distances of a point from the centres. This example was given before 

 Section A at York last year by Professor A. W. Eiiiclcer.' 



Straight Lines and Circles. — Here again our first reference must be to 



Stokes. In his paper ' On the critical values of the sums of periodic 



scries' he finds the velocity potential for the fluid inside a rotating sector 



of a circle of angle 2 a in the two forms of an infinite series and of a 



definite integral, and expresses in the same two forms the moment of 



inertia of the equivalent solid. The square of the radius of gyration is 



16 f* tanh ax 



— — — ^ — T^ dx. For special values of the angle «, the velocity poten- 



Trajo x{x' + 4>)-' 



tial and stream functions admit of finite expression in terms of logarithmic 



and circular functions. The semicircle is the simplest,^ next comes the 



quadrant of a circle, and a sector of 60°, the two last given by Greenhill,^ 



who has investigated the case of the sector very fully. In this paper 



the case when the angle is any sub-multiple of two right angles is also 



considered. The expressions obtained are naturally rather complicated, 



but they are finite and in terms of circular and logarithmic functions. 



For the two particular cases of the semicircle and quadrant he shows 



that the ratios of the squares of the radii of gyration of the equivalent 



solid, and the solidified fluid, are IG/tt^ — 1, and (16 log 2)1^^—^, 



respectively. He again takes up the question in a later paper'' (1880), 



and obtains a finite expression for the functions when the angle of the 



sector is commensurable with two right angles. The square of the ratio 



of the radius of gyration of the equivalent solid to the radius of the 



circle is in this case 



where ^ («) = ^ log r (x) . 



When sources and sinks exist inside a sector the motion may easily be 

 determined by means of what is already known for the space between 

 two lines. The position of rest of a vortex inside a sector has been de- 

 termined by Lewis.^ It lies on the line bisecting the angle at a distance 

 from it = { V (4)2-2 ^ i) _ 2n] ^'2" times the radius, the angle of the sector 

 being x/^i. The general motion both for vortices outside and inside a 

 circle — either for a single vortex in a sector, or symmeti'ical vortices, 

 without straight boundaries, is given by Greenhill ^ with figures illustrating 

 the paths for two, four, and six vortices respectively. 



The same author has also discussed the motion in the space bounded 

 by two concentric circles and two radii. The values of f and -^ for a 



' ' On a problem in stream lines,' Brit. Assoc. Hep. 1881, p. 554. 

 - 'Fluid motion in a rotating semi-circular cvlinder/ Mess. Math. viii. p. •12 

 (1878). 



3 'Fluid motion in a rotating quadrantal cylinder,' Mess. Math. viii,' p. 89 (1877). 



* 'On the motion of a frictionlcss liqiiid in a rotating sector,' Mess. Math. x. p. 83. 



* 'Some cases of vortex motion,' 3Iess. Math. Ix. p. 93 (1879). 

 " 'Plane vortex motion,' Quart. Jour. xv. p. 10 (1877). 



