102, 103.] SOLIDS OF REVOLUTION. 117 



103. A method similar to that of Art. 88 has been employed 

 by Rankine* to discover forms of solids of revolution (about x 

 say) for which &amp;lt;/&amp;gt; 1 is known. When such a solid moves parallel to 

 its axis the motion generated in the fluid takes place in a series 

 of planes through that axis and is the same in each such plane. 

 In all cases of motion of this kind there exists a stream-function 

 analogous to that of Chapter IV. If we take in any plane 

 through Ox two points A and P, A fixed and P variable, and 

 consider the annular surface generated by tbe revolution about x 

 of any line AP, it is plain that the quantity of fluid which in 

 unit time crosses this surface is a function of the position of P, 

 i. e. it is a function of x and -cr, where TX denotes the distance of 

 P from Ox. Let this function be denoted by 27n/r. The curves 

 ijr = const, are evidently stream-lines, so that ty may be called the 

 * stream-function. If P be a point infinitely close to P in the 

 above-mentioned plane, we have from the definition, of A|T 



fluid velocity normal to PP = ^ -pp r , 



and thence, taking PP parallel, first to *r, then to x, 

 1 d-^r 1 dty 



where u, v are the components of fluid, velocity parallel to x and ir 

 respectively. 



For the case of the sphere, treated in Art. 102, we readily find, 

 by comparison of (12) and (13) 



So far we have not assumed the motion to be irrotational. 

 The condition that it should be so, is 



du dv _ - 

 dtz dx 

 which reduces to 



. . ^ = (15). 



Phil. Trans. 1871. 



