440 Prof. Challis on the Principles uf Hydrodynamics. 



of the fluid so act on each other, that a motion of this kind 

 results ? The absurdity to which, as ah'eady stated, this suppo- 

 sition conducts, proves that it is not allowable. Again, motion 

 which tends to or from a fixed centre, and is a function of the 

 thstance from the centre, is rectilinear motion, and satisfies the 

 criterion of iutegrability of udx + vdy + wdz. The absurdity, how- 

 ever, to which the supposition of such motion conducts proves 

 that this is not the kind of motion resulting from the mutual 

 action of the parts of the fluid. Neither can it be motion tend- 

 ing to or from focal lines ; for if this were the general law, no 

 absurdity would result in the particular case of motion tending 

 to or from a centre. Thus the absurd results above cited are 

 extremely important, as excluding from our consideration the 

 kinds of motion just mentioned. 



(2.) It remains to consider the case of an axis of rectilinear 

 motion. The general integrability oS udx + vdy -\- wdz is in this 

 case only satisfied by the motion along, or immediately conti- 

 guous to, the axis, the motion at all other points being curvi- 

 linear. For the purpose of tracing the consequences of this 

 supposition, let 



, df ,df j,dd) 



"=^;^' ^^'^^^ ^=^^' 



/being a function of x and y only, and <p a function of z and t 

 only. Further, let the function /be such, that where a? =0 and 

 y = 0, we have 



f^l ^/=o ^=0. 

 '' ' dx ' dy 



It is clear that on these suppositions udx + vdy -f ivdz is integrable, 

 and that the axis of 5- is a hue of motion. If no contradictory 

 results, similar to those before indicated, be arrived at by tracing 

 the consequences of the above suppositions, the motion due to the 

 action of the parts of the fluid on each other must be of the kind 

 here assumed, because it is certain, a priori, that that motion is 

 unique and perfectly definite. 



Now as a first consequence of our hypothesis, we have 



{d.f(f))=udx + vdy + ivdz (a) 



Combining with this equality the general equation of Proposition 

 IV., viz. 



dp d.pu d.pv d.pw _ „ 



di'^i^^W ~dr-^' • • • (^) 



and that which the general equation of Proposition V. becomes 

 when there are no impressed forces, viz. 



