residue theorem used even in the case of rotational motion. For example a circular 

 cylinder exposed to a stream V on which is superposed uniform shear flow of vorticity 

 co undergoes the lift Trpd 2 toV. 



Just as the theorem of Stokes is fundamental in "solid flow," if I may use that 

 expression, so is the form which the theorem assumes in two dimensions in terms of z 

 and z a powerful tool. This I have named the complex Stokes' theorem [8]. The 

 complex Stokes' theorem refers to a plane area S bounded by a closed curve C and 

 states that 



f f d - f 



f(z,z)dz = 2i\ —dS. (15) 



Jc Js dz 



Cauchy's theorem is a particular case, namely when df/dz — 0. 



One immediate and important application is the calculation of the kinetic 

 energy [9] of liquid in irrotational motion, with its application to virtual mass. Thus 



(22) 



where w = <p + iib is the complex potential. If the region 5 is multiply-connected, 

 suitable barriers must be introduced as part of the boundary C. Note that in virtue of 

 the remarks made shortly ago the integral round the boundary can be evaluated by 

 the residue theorem since z is a function of z. 



The generalization of the complex variable to three-dimensions leads to Hamil- 

 ton's quaternions. Alan Rose [10] defines a stream function \b{x, y, z\ f, -q, £) as the 

 flux across the triangle formed by the origin, the point A(x, y, z) and the point 

 B(x + £, y + 77, z + £)• If we define the vector 



/ d^ diA 3$ \ 

 * = OAi, H *,) = — , — , — (23) 



\ d£ drj 6f /s_„-r-o, 



the velocity is 



q = - V A *. (24) 



If the motion is axisymmetric and irrotational with velocity potential <p, the 

 function 



<P + *Vi + JH + ty 3 (25) 



satisfies the condition for it to be a right-regular quaternion function of the quaternion 

 variable 



w + ix + jy + kz (26) 



where w is an imagined coordinate whose axis is perpendicular to the axes of x, y, z 

 and i, j, k are Hamilton's unit vectors. 



Thus the theory of analytic quaternion functions is in principle available. 



In this way, for example, it is possible to deduce the flow past a sphere in terms 

 of the quaternion variable by a method entirely analogous to that for deducing the 

 flow past a circle in terms of the complex variable. 



Here then is a method which may well merit further investigation. 



Let us now turn to a problem which is important from the naval standpoint; 

 the problem of virtual mass. To take the simplest case when a body of mass M moves 



