4850.] THE VELOCITY-POTENTIAL DETERMINATE. 45 



theory of Attractions, was first given by Gauss (I c. Art. 46). 

 Another method of proof, purely analytical in form, and applicable 

 to theorems (/3) and (7) as well, was given by Thomson* in 1848. 

 As this method is not free from difficulty, it is not reproduced 

 here. It is however perfectly easy to shew analytically that in 

 theorems (a) and (7) of Art. 48 &amp;lt;j&amp;gt; is completely determinate, i.e. 

 that there is only one function satisfying the conditions stated, 

 and that in (/3) &amp;lt;/&amp;gt; is determinate save as to an additive constant. 



In the first place, if &amp;lt; 1? &amp;lt; 2 , &amp;lt; 3 , &c. be velocity-potentials of 

 possible states of motion throughout any given region, then 



where A^ A z , A 3 , &c. are any constants, is the velocity-potential 

 of a possible state of motion (throughout the region). This 

 follows from the linearity of (10). 



Now, if possible, let there be two single-valued functions 

 t , &amp;lt;/&amp;gt; 2 , each satisfying the conditions of (a) with respect to any 

 region. Then 0j &amp;lt;/&amp;gt; 2 satisfies (10) throughout this region and is 

 zero at every point of the boundary. It is therefore, by Art. 47, 

 zero throughout; i.e., &amp;lt;p l} &amp;lt;/&amp;gt; 2 are identical. 



Again, if it be possible, let ^, &amp;lt;f&amp;gt; 2 be two single- valued functions 

 each satisfying the conditions of (/:?) with respect to any region. 

 Then &amp;lt;^ l c/&amp;gt; a satisfies (10) throughout this region, and makes 



all over the boundary. Hence by Art. 47 we have (f&amp;gt; l (/&amp;gt; 2 

 constant throughout the region ; and therefore the motion, which 

 is determined by the derivatives of &amp;lt;/&amp;gt;, is the same in each case. 



Lastly, if &amp;lt;p lt &amp;lt; 2 be two single-valued functions satisfying the 

 conditions of (7), it is seen in the same way that we must have 



&-*, = o. 



50. A class of cases of great importance, but not strictly in 

 cluded in the scope of the foregoing theorems, are those where the 

 region occupied by the liquid extends to infinity, but is bounded 

 internally by one or more closed surfaces. We assume, for the 



* See Reprint, Article xin. Also Thomson and Tait, Natural Philosophy, 

 Appendix A (d). This method is often attributed by German writers to Dirichlet. 



