40 BEPOET— 1882. 



body are varied in the same way as if the solid were to have one of its 

 possible motions. 



On the same page it should be noted that the theory of similitude 

 given by Helmholtz had already, more than twenty years before, been 

 stated by Stokes,' though not so fully as to include different coefficients 

 of gliding over the surfaces of bodies immersed. The applications by 

 Helmholtz are new. 



On page 80, line 20, for ' its i-ate of variation,' I'ead, ' the jrate of 

 variation of the energy.' 



The following letters have throughout, nnless specially not%|Bd, the 

 meanings here given, viz. : — 



Where on denotes the mass of a body, in' denotes the mass of the 

 fluid displaced by it. 



p denotes the density of the fluid. 



(j) „ ,, velocity potential. 



if/ „ „ stream function. 



fj.' ,, „ coefficient of viscosity. 



H „ ,, kinematic coefficient of viscosity = /''/p- 



a. Motion in Ttvo Dimensions. 



Sources and SinJcs. — The simplest motion possible is that where fluid 

 moves in an infinite plane, streaming from certain points (sources) and 

 into others (sinks). Its importance consists in this, that all potential 

 functions can be considered as due to certain distributions of sources or 

 sinks at definite points, or along certain lines and surfaces, as has been 

 shown by Stokes.^ Regarded from this point of view they have been 

 called the ' Green's functions of the given distribution of matter.' Many 

 examples will occur in the succeeding pages of their application. W. R. 

 Smith ^ has developed some of the general properties of the stream-linos 

 and equipotential lines for two dimensional motion when the number of 

 singular points is finite and all are of the same magnitude. When the 

 system is complete, i.e. when the numbers of sources and sinks are equal, 

 the degrees of both the stream-lines and equipotential lines are equal to 

 the whole number of singular points. When the numbers are unequal 

 this is still true for the stream-lines, but the degree of the equipotential 

 line is double the greater number. The general nature of the lines is 

 clearly different, according as the system is complete or not. In the 

 former case one stream-line goes to infinity, and is, at most, of a degree 

 one less than the number of singular points, whilst if the system be not 

 complete, eveiy complete stream-line has a number of asymptotes equal 

 to the difference of the numbers of sources and sinks. More jjarticularly 

 he considers the cases of two, three, and four singular points, and gives 

 figures when they are at the corners of a rectangle and of a regular 

 trapezium. Cases of the same kind have also been noticed by Kirchhoff "^ 



' Cami. Phil. Trans, ix. 



^ ' On the internal distribution of matter which shall produce a given potential 

 at the surfaces of a gravitating mass,' Proc. Roy. Soc, xv., p. 482 ; and Phil. Mag., 

 xxxiv. p. 235 (1867). 



^ ' On the flow of electricity in conducting surfaces,' Proc. Roy. Soc. Edin., vii. 

 p. 79. 1870. 



* ' Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbeson- 

 dere durch eine kreisformige,' Pogg. Ann. Ixiv. 15, 497, or Gemni, Ahhantl, p. 1. 



