132 THE MECHANICS OF THE EAKTh's ATMOSPHERE. 



be SO bouuded that, witliiu it, no branch of oj merges into another ; 

 such a branch, therefore, represents a possible mode of fluid motion. 

 The desired object will be attained when the region of co is appropri- 

 ately bounded. 



In reference to the boundary of the region of ai it is recognized, first, 

 that it is a line that returns into itself and without cutting itself and 

 that consists of parts for which //• has a constant value and of parts 

 for which q) has an indefinitely large positive or an indefinitely large 

 negative value. 



Within the region of cj,/(&?) is a single- valued function of co. If we 

 had adopted an expression for/(Gj) that represented a many- valued 

 function, then at its cusp point should start the sections for which w 

 has a constant value. 



Furthermore y/ J\oj)f{ay) — l should also be made a single-valued 

 function of co. in that through those points for which /(CL)) = i 1, the sec- 

 tions pass for which //• has a constant vahie. For any point of the 

 region of co the sign of the radical quantity is still at our disposal. If 

 points occur for which /(a?) is infinite or Infinitely great,* then for one 

 of these points we may make 



^ff{co)f(co)-\ = ^f{co) 



and assume that this equation holds good for them all. 



It is further assumed that the function /(&?) is only infinite at its cusp 

 points if it is so anywhere, and even here it is infinite only in such a w ay 

 that if f{oo(s) is infinite then {co—con)f{co) approximates to zero when co 

 has a value approximating that of co^. 



Within the designated region of co therefore z is a single-valued 

 function of this variable and such that it is never infinite. 



Now consider CcD as a function of ;;. The region of s that corresponds 



to the adopted region of co does not extend through infinity, and is 



bounded by a line that returns into itself and which is made up of the 



lines whose equations are cp= — x> and (^= + 00 and of stream lines; a 



certain portion of the latter cm be considered as a free boundary of 



the moving fluid, the other part can be considered as a fixed wall. 



Within this region of z, co has no cusp point, since at no point of it 



dz 

 does^ become zero. Therefore under the condition that the boundary 



of the region of z shall not intersect itself, co becomes within that 

 region a single valued function of z. 



This function of z is completely determined as soon as one has found 

 a single value of z corresponding to a given value of co. 



(I.) An example that constitutes a generalization of the case treated 

 of by Helmholtz is obtained if we put 



/(Gj)=/v-+e-<" 



* By infinite, I designate the reciprocal of zero, but by infinitely great, the recipro- 

 cal of an infinitely small quantity. 



