Attrajstive and Repulsive Forces. 201 



cause these rectilinear motions are produced under arbitrary 

 conditions. Besides, it does not appear that for such motions 

 -x/r has the character of a maximum or minimum. The vahdity 

 of the foregoing argument is confirmed by the fact, which I 

 think I have on previous occasions sufficiently certified, that the 

 two cases of rectilinear motion just mentioned cannot be mathe- 

 matically treated with success till the circumstances and laws of 

 the rectilinear motion now under consideration have been as- 

 certained. 



26. It is further to be noticed that, as the indication of recti- 

 linear motion was arrived at by employing one general equation 

 without reference to the other tw^o, the result is not of general 

 application, and we are not compelled to infer from it that the 

 motion is always and everywhere rectilinear. Yet, according to 

 the rule already laid down, the indication must be significant, 

 and cannot without error be left out of consideration on proceed- 

 ing to draw inferences from the other two equations. Accord- 

 ingly I propose the following course of reasoning. 



As the motion is not exclusively rectilinear, let it be supposed 

 to take place in part along a straight line which relatively to the 

 density and the rest of the motion is an axis. This hypothesis 

 may be analytically expressed by first supposing the axis of the 

 motion to coincide with the axis of 2", and then assuming that 



( f/T^) = (d ./(/)) = udx -}- vdy -f ivdz, 

 /being a function of x and y onl}^, and <^ a function of -s- and t 

 only. The complete exhibition of the present argument would 

 require to be given next an investigation of the consequences to 

 which this supposition leads when the second and third general 

 equations are taken into account. As this investigation is too 

 long for insertion here, I can only refer to it as given in the Phi- 

 losophical Magazine for August 1862, and in fuller detail in the 

 proofs of Propositions XI. to XVII. contained in pages 201-240 

 of the volume I have so frequently cited already. The researches 

 in this portion of my work, which are of a peculiar and novel 

 character, lead to certain results which, according to my view?, 

 are indispensable for the progress of analytical physics. This 

 rem^ark will receive illustration from the use I now proceed to 

 make of these results. 



27. The mathematical consequences that are deduced from 

 the above hypothesis relative to the composition of the diflPeren- 

 tial {dy\r), after taking into account the second and third general 

 equations, are found to involve no contradictions, are definite 

 and unique as regards both the motion and the density of the 

 fluid, and account at once for certain observed facts which are 

 due to essential properties of the fluid apart from arbitrary con- 

 ditions, such as the fact of uniform propagation, and that of co- 



