CONTENTS. 



CHAPTER III. 



IRROTATIONAL MOTION. 



ART. PAGE 



31. Analysis of the motion of a fluid element .... 33 



32, 33. Flow ' and * Circulation.' Stokes' Theorem .... 35 



34. Constancy of circulation in a moving circuit . . . .38 



35, 36. Irrotational motion in simply-connected spaces ; </> single- 



valued 39 



37-39. Case of an incompressible fluid ; tubes of flow. < cannot 

 be a maximum or minimum. Mean value of over a 

 spherical surface . .40 



40, 41. Conditions of determinateness of < 44 



42-46. Green's theorem. Dynamical interpretation. Formula for 

 kinetic energy. Lord Kelvin's theorem of minimum 

 energy 47 



47-51. Multiply-connected regions. Irrotational motion ; cyclic 



constants 53 



52. Conditions of determinateness for the motion of an incom- 

 pressible fluid in a cyclic region 58 



53-55. Lord Kelvin's extension of Green's theorem ; dynamical in- 

 terpretation. Energy of an irrotationally moving liquid 

 in a cyclic space ........ 60 



56-58. ' Sources ' and ' sinks.' Double-sources. Surface-distributions 



of simple and double sources 63 



CHAPTER IV. 



MOTION OF A LIQUID IN TWO DIMENSIONS. 



59. Stream-function. . . . . . . . . .69 



60, 61. Irrotational motion. Kinetic energy . . . . .71 



62. Connection with the theory of the complex variable. Con- 



jugate functions. . . . . . . . .73 



63, 64. Simple types of motion, acyclic and cyclic .... 78 



65. Inverse formulae. Examples . . . . . . .81 



67-70. General formulae; Fourier's method. Motion of a cylinder, 



without and with circulation of the fluid round it . 84 



71. Inverse methods. Motion due to translation of a solid. 



Elliptic cylinder. Flow past an oblique lamina ; couple- 

 resultant of fluid pressures 91 



72. Motion due to a rotating solid. Rotating prismatic vessel ; 



cases where the section is an ellipse, a triangle, or a 

 circular sector. Rotating elliptic cylinder in infinite 

 liquid 95 



