RECENT PEOGEESS IN HYDRODYNAMICS. 75 



question was takeu up by Boltzmann' later, wlio has considered the case 

 of non-cii-cular section. He notices that Kirchhoff's analogy is not 

 exact, as the forces due to the motion of the rings vanish in the case of 

 fluid motion. 



Kirchhoff's investigation of the solid of revolution was completed 

 in 1877 by Kopcke,^ who did for the general case of a solid of revolution 

 what Kirchhoft' had done for Thomson and Tait's solution. He succeeded 

 in expressing the elements of the motion and the position at any moment 

 by rueans of the elliptic and 9 functions, using a quadric transformation 

 to reduce the functions in «, the velocity along the axis, to the canonical 

 form. In this also two chief cases occur distinguished as in the simpler 

 case mentioned above. The same end was also attained by a somewhat 

 different process by Greeuhill.^ 



In the case of any solid whatever, we obtain at once three integrals in 

 the form of constant energy, constant impulsive force, and constant 

 impulsive couple. Clebsch noticed that if a fourth integral could be 

 obtained, a fifth could be at once deduced by the principle of the last 

 multiplier ; and the last integral, giving the value of t, be then found by 

 quadratures. In the ' Mathematische Annalen '* for 1871, soon after 

 Kirchhoff's paper, he set himself the problem to discover when the equa- 

 tions admitted («) of a linear integral, (/3) of a quadric integral not com- 

 pounded of the first three integrals. Instead of the velocities, he takes 

 the momenta for dependent variables. Writing x-^, x^, x-^, for the 

 component impulsive forces of the motion and y,, 1/2, ijs, for the com- 

 ponent impulsive couples, the energy can be expressed in the form Tj + 

 T2 + T3 where T,, T3 are quadric functions of the (x) and (y) respec- 

 tively, and T 2 contains only products of an x into a y, which he proves 

 may by a proper choice of the origin be such that the coefficients of X; yj 

 and Xj yi are the same. For a linear integral the condition found is that 

 T must be expressible in the form ^ — 



a(xi^ + X2^) + aix^'^ + b (x^y^ + x.2 yo) + fix-^ys + ciy^^ + y^^) 



when 2/3:= const, is an integral. This includes Kirchhoff's case and 

 Thomson's isotropic helicoid. For a quadric integral three cases appear, 

 one of which reduces to the square of the above ; the other two are that 

 T must be expressible in either of the forms 



Ti + A (a!i2/i + a;2 2/2 + ^s^s) + fJ- (2/1^+2/2^ + 3/3^) (") 

 or a^x^"^ + a^x^ ^- a^x-^ ■\- l^y^ -H l^y^ + Izy^ + X {x^y^ 



4- x^y<2, 4- 0332/3). (/3) 



where o-i-az ^ ^■' ~ ^1 4. ^iJZ±2_o 



&i ^2 *3 



when the integrals are respectively 



for (a) ^T,(?/) — A = const. 



' ' Ueber die Druckkrafte, welclie auf Einge wirksam sind, die in bewegte Fliissig- 

 keit tanchen,' Borch., Ixxiii. p. 111. 



- ' Zur Discussion der Bewegnng eines Eotationskorpers in einer Fliissigkeit,' 

 Math. Annalen., xii. p. 387. 



' ' Motion of an ellipsoid in liquid,' Quart. Jour. Math., xvi. p. 242. 



* ' Ueber die Beweguug eines Korpers in einer Fliissigkeit,' Math. Ann., iii. 

 p. 238. 



* In other words, the solid mixst be similar to itself turned through one right 

 angle. 



