170 VORTEX MOTION. [CHAP. VI. 



142. The kinds of fluid motion now under consideration 

 come under the class for which a stream-function ^ was shewn, 

 in Art. 103, to exist. By the definition of that article, we have 



2-m/r = total flux through the circle whose co-ordinates are x, -or, 

 = ff(lu + mv + nw) dS, 



where the integration extends over any surface bounded by that 

 circle. Recalling the expressions (6) for u, v, w, from which P is 

 now to be omitted, we have by the theorem of Art. 40, 



27rf = l(Ldx + Mdy + Ndz), 

 the integration here being taken round the circle, or, by (47), 



V = w# ........................... (50). 



The formula (28) for the kinetic energy may now be written 



(51). 



143. Let us take the case of a single circular vortex of 

 strength m. At all points of its infinitely small section the 

 modulus k of the elliptic integrals in the value of 8 is nearly 

 equal to unity. In this case we have* 



4 

 ^i = lo g^&amp;gt; ^=2^ 



approximately, where k denotes the complementary modulus 

 & 2 ), so that in our case 





nearly, if S denote the distance between the infinitely near points 

 (x t -BT), (x, r ). Hence at points within the substance of the 

 vortex the value of S, and therefore by (50) also of ty, is of the 

 order mloge, where e is a small linear magnitude comparable 

 with the dimensions of the section. The velocity at the same 

 point, depending (Art. 103) on the differential coefficient of ^, 



777 



will be of the order . 

 e 



* See Cayley, Elliptic Functions, Art. 72, and Maxwell, Electricity and 

 Magnetism, Arts. 704, 705. 



