1891.] 



Fluid Motions in two dimensions. 



201 



so that 



XI = loo- \B 



'i + V(i-«0" 



Ai 



= log 



dz 

 dw' 



- V(l - w) 



and since II = when w = 1, and the argument of dz increases by 

 (it — a) as w goes through z%xo, we find 



dz 

 dw 



l + vXi-w) 



1 - V(l - w) 



1 — a/7r 



.(45). 



16. With our choice of constants the length of the part 

 between the points corresponding to w = and w = 1 is 



J" 



Jo 



1 + V(l - wj 

 1 - V(l - w). 



1 - ajir 



dw, 



putting w = sin 2 and tan \6 = x, ajir = n, we transform this into 



(I-* 2 ) 



dx. 



.(46), 



o (1+tf 2 ) 3 



and when a is an exact submultiple of tt, n is an integer and we 

 shall be able to evaluate the integral. 



The pressure on the part between the same two points is 



J o 



dw 

 dz 



dz 

 dw 



dw, 



or, making the same transformations as before, 



4/j 



^ 



') (1 -<*) s t 



n o (X-vb 



.(47). 



o (l+<> 3 



The above might be applied to find the pressure on the rudder 

 of a ship when turned obliquely to the length of the ship. The 

 average pressure, the centre of pressure, and the moment of the 

 fluid pressures can all be expressed by means of definite integrals 

 of similar form to the above. If it were worth while tables might 

 be constructed giving the values of these quantities for any given 

 inclination of the plane of the rudder to the longitudinal plane of 

 the ship. As however the motion here considered is in two 

 dimensions it is unlikely that the formulge would yield any result 

 of use in Navigation. 



(4) On thin rotating isotropic disks. 

 Fellow of King's College. 



By C. Chree, M.A., 



The following solution might be shortened by assuming certain 

 results from a previous paper in the Transactions 1 . As the 

 subject appears, however, to be one of considerable practical 



1 Vol. xiv. p. 328 et seq. 



