118 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



The differential equation satisfied by &amp;lt;j&amp;gt; now assumes the form 



idi&amp;gt;_ 



- 



This may be derived by transformation from (1), by writing 

 y = TV cos 0, z = TX sin 0, 



and remembering that &amp;lt; is now independent of 6, or by repeating 

 the investigation of Art. 12, taking, instead of the elementary 

 volume dxdydz there considered, the annular space generated by 

 the revolution about x of the rectangle dxdtx. It appears that &amp;lt; 

 and ^r are not, as they were in Chapter IV., interchangeable. 



104. Rankine s procedure is then as follows. Supposing the 

 solid to move parallel to its axis with velocity V, we have at all 

 points of a section of its surface made by a plane through Ox t 



- = velocity in direction of normal 



-, 



as 

 ds denoting an element of the said section. 



Integrating along the section, we find 



^ = 1JV + const ......... ... ............ ........ (17). 



If in this equation we substitute any value of ty satisfying (15), 

 we obtain the equation of the meridional section of a series of 

 solids of revolution, any one of which would when moving parallel 

 to its axis produce the system of stream-lines corresponding to the 

 assumed value of ty. 



In this way may be verified the value (14) of ^r for the case of 

 a sphere. 



Dynamical Investigations. 



105. The second part of the problem proposed in Art. 99 

 is the determination of the effect of the fluid pressure on the 



