of certain Stream-Lines. 283 



Part I. — On some Exponential Stream-Lines in two 

 Dimensions. 



2. It is well known that amongst the functions which satisfy 

 the conditions of liquid motion in two dimensions, are compre- 

 hended all those of the form 



2/ + Z . e^cos ax. 

 Such functions as the above obviously represent curves consist- 

 ing of an endless series of repetitions of the same figure ; and 

 many of those curves resemble the profiles of waves. 



3. The first part of the investigation consists of a discussion 

 of the properties of the curves represented by the simplest of 

 those exponential stream -line functions, viz. 



b — y — e~ y cosa? (I.) 



By giving to b a set of values in arithmetical progression, this 

 function is made to represent a set of stream-lines, dividing an 

 indefinitely extended plane layer of liquid into a series of curved 

 streams of equal flow. Each of those stream-lines consists of an 

 endless series of repetitions of the same figure, the length parallel 

 to x of each repetition being 27r; and each repetition consists of 

 a pair of symmetrical halves. 



4. The graphic construction of those stream-lines is very 

 easy, by the aid of a general method of constructing curves first 

 used by Professor Clerk Maxwell, and applied by the present 

 author to stream-lines in the previous investigation already 

 referred to. 



Draw a series of straight lines parallel to x, and having for 

 their equation 



y = m } 



— the values of m being in arithmetical progression, positive and 

 negative, with a fraction for their common difference, which 

 should be the smaller the more accurate the drawing is to be. 

 Then draw a series of curves of hyperbolic-logarithmic cosines, 

 having for their equation 



e~ y cosx = 7n' } 



— the values of m!, positive and negative, forming an arithmetical 

 progression, and having the same common difference with those 

 of m. The curves with positive values of m' lie between x = 



and x= —; those with negative values between x= — and#=7r; 

 £ Z 



and the straight line parallel to y, at -, is an asymptote to them 



all. One and the same mould serves to trace all those curves ; 

 for they differ only in the maximum value of y, which is 



U2 



