568 BEPOET — 1888. 



2. In the Secondary and Tertiary the motion is essentially without change of 

 density, and in each of them we naturally, therefore, take an incompressible fluid as 

 the substance. The motion in the Primary we arbitrarily restrict by taking its fluid 

 also as incompressible. 



3. Helmholtz first solved the problem — Given the spin in any case of liquid 

 motion, to find the motion. His solution consists in finding the potentials of three 

 ideal distributions of gravitational matter having densities respectively equal to 

 l/47r of the rectangidar components of the given spin; and, regarding for a 

 moment these potentials as rectangular components of velocity in a case of liquid 

 motion, taking the spin in this motion as the velocity in the required motion. 

 Applying this solution to find the velocity in our Secondary from the velocity in our 

 Tertiary, we see that the three velocity components in our Primary are the poten- 

 tials of three ideal distributions of gravitational matter, having their densities 

 respectively equal to Ij-iir of the three velocity components of our Tertiary. 

 This proposition is proved in a moment,' in § 5 below, by expressing the velocity 

 components of our Tertiary in terms of those of our Secondary, and those of our 

 Secondary in terms of those of our Primary, and then eliminating the velocity com- 

 ponents of Secondary so as to have those of Tertiary directly in terms of those of 

 Primary. 



4. Consider now, in a fixed solid or solids of no magnetic susceptibility, any 

 case of electric motion in which there is no change of electrification, and therefore 

 no incomplete electric circuit ; or, which is the same, any case of electric motion in 

 which the distribution of electric current agrees with the distribution of velocity in 

 a case of liquid motion. Let this case, with velocity of liquid numerically equal to 

 4tt times the electric current density, be our Tertiary. The velocity in our corre- 

 sponding Secondary is then the magnetic force of the electric current system;" and 

 the velocity in our Primary is what Maxwell ^ has well called the ' electro-magnetic 

 momentum at any point ' of the electric current system ; and the rate of decrease 

 per unit of time, of any component of this last velocity at any point, is the corre- 

 sponding component of electro-motive force, due to electro-magnetic induction of 

 the electric current system when it experiences any change. This electro-motive 

 force, combined with the electrostatic force, if there is any, constitutes the whole 

 electro-motive force at any point of the sjstem. Hence by Ohm's law each com- 

 ponent of electric current at any point is equal to the electric conductivity multi- 

 plied into the sum of the corresponding component of electrostatic force, and the 

 rate of decrease per unit of time of the corresponding component of velocity of 

 liquid in our Primary. 



5. To express all this in symbols let (Mj, v^, iv,), (m.^, i\, w.,), and (u^, v^, w^) 

 denote rectangular components of the velocity at time t, and point (.r, i/, z) of our 

 Primary, Secondary, and Tertiary. We have (§ 1)— 



d7V, dv, du, dw, dv, du, ztv 



' dy dz' ^ dz dx ' dx dy ^ ' 



diVo dv.^ du^ diL\ _dOo du„ ,n\ 



dy dz dz dx dx dy 



Eliminating m^, v^, w^ from (2) by (1), we find — • 



d (du. dv, . dw,\ (d-u, . dru. dru.\ «„ /ox 



dx\dx dy dz ' \dx- dy^ dz- / 



But by our assumption (§ 2) of incompressibility in the Primary — 



^ + ^1 + ^1 = (4) 



dx dy dz 



Hence (3) becomes — 



M3=-V'"i. «'3=-VX. »'3=-V'^«'i ... (5) 



* From Poisson's well-known elementary theorem v'V'= -- itrp. 

 - Electrostatics and Mayiietism , § 517 (postscript) (c). 



• Electricity and Magrietism, § 604. 



u. 



