194 Prof. Challis on the Hydrodynamical Theory of 



9. Resuming tlie equation (a), and putting it under the form 



dt ^ dt ^^^^' 



there will be two cases for consideration : first_, that for which ^jr 

 is independent of t, in which case [iidx'] may, from what is said 

 in the preceding paragraph^ either be an exact differential, or be 

 integrable by a factor; and secondly, that in which -v/r varies 

 with the time and [udw] is necessarily an exact differential. 



10. Taking the case in which yjr does not contain t, we have 



-^ = 0, so that {d-^) = 'xj^t)dt. In order that this result may be 



consistent with the condition that -v/r is independent of ^, we must 

 hav^e '^[t)=Q, an arbitrary constant ; and then, since dt may be 

 considered constant, it will follow that [d^\r)j which is the change 

 of -v//^ in passing at a given instant from any point to a contiguous 

 point, is an arbitrary infinitesimal quantity. Being of arbitrary 

 value we may suppose it to vanish, or that {d-\\r) — 0. Since this 

 is the differential equation of a surface of displacement, the pre- 

 ceding argument has shown that such a surface can exist, con- 

 sistently with the principle of geometrical continuity, in every 

 case of steady motion, whether [udxl be an exact differential or 

 be integrable by a factor. 



11. It will be now proper to state that in the proposed hydro- 

 dynamical theory of the physical forces, certain of these forces are 

 referable to pressure of the aether in steady motion, while the 

 remainder are accounted for by pressure accompanying its un- 

 steady or vibratory motions. Two classes of forces, known ex- 

 perimentally, are found to correspond to two kinds of motion of 

 a fluid which have been ascertained by the aid of mathematics. 

 The forces which correspond to steady motions of the aether will 

 be first considered. 



12. The way in which the particular cases of steady motion 

 for which \udx] i% integrable by a factor account for the conduc- 

 tion of galvanic currents along slender wires has already been 

 sufficiently referred to in art. 8 of this communication, and need 

 not be discussed here. 



13. The forces which, according to the theory, are ascribable 

 to pressures of the sether in the cases of steady motion for which 

 [udx] is an exact differential, are magnetic, galvanic, and electric 

 attractions and repulsions. It is assumed to be a necessary con- 

 dition of the existence andmaintcDance of the appropriate steady 

 motions, that a gradation of atomic density should exist in the 

 interiors of the attracting or repelhng substances. In arts. 4 to 

 9 of the Theory of Magnetism in the June Number I have pro- 



