which has at every point of the boundary its rate of variation 



44 IRROTATIONAL MOTION. [CHAP. III. 



boundary, the arbitrarily given surface-value must, with its rate 

 of variation from point to point, be finite and continuous. 



Again, let us consider a mass of liquid initially at rest and 

 enveloped by a perfectly smooth flexible membrane which it just 

 fills ; and let us suppose that every point of the membrane is 



suddenly moved with a given normal velocity rr- , which must of 



course satisfy the condition II ^ dS = 0, where the integration 



extends over the whole membrane. Since&quot; some definite motion of 

 the mass must ensue, we are led, on the same kind of evidence as 

 before, to the following analytical theorem ; 



(/3) There exists a single-valued function ^&amp;gt; which satisfies 

 the equation y 2 = at every point of a given region, which is with 

 its first derivatives finite and continuous throughout that region, and 







in the direction of the normal equal to a given arbitrary value, 

 subject to the condition 1 1 -^ dS = 0. If the finiteness and con 

 tinuity of $ and its derivatives hold right up to the boundary, the 



given surface-values of ~ must be continuous. See Art. 84. 

 dn 



(7) Lastly, combining the two modes of genesis of motion 

 described above, we are led to enunciate the theorem that a single- 

 valued function exists which satisfies y 2( /&amp;gt; = at all points of a 

 given region, which is with its first derivatives finite and continuous 

 throughout the region, and which has a given arbitrary value over 

 one part of the boundary and gives an assigned arbitrary value of 



- over the rest of the boundary. 

 an 



49. The physical considerations adduced in support of the above 

 theorems are due to Thomson*. The theorem (a) was originally 

 stated by Green *f, and based by him on electrical considerations. 

 An analytical proof, couched however in the language of the 



* Reprint of Papers on Electrostatics and Magnetism, Article xxviu. 

 t An essay on Electricity and Magnetism (1828), 6. 



