28 Prof. Challis.cn Newton's 



tions, which suffice for determining the seven unknown quan- 

 tities yfr, X, u, v, w, p, p as functions of a, y, 0, and t. (See 

 another investigation of the same equation in pp. 174 and 

 175 of the work already cited.) 



As the equation (A) serves for determining A, as a function 

 of x, y, z, and t, and as I have not been able to discover any 

 fault in the reasoning by which it was reached, I regard it as 

 giving proof of the reality of motion for which udx + vdy+ wdz 

 is integrable by a factor. In short ; reasons may be adduced 



for concluding that the factor - is applicable to cases in which 



the motion is not distinctively that of a fluid, but such as a fluid 

 is capable of if conceived to be composed of infinitely small 

 parts that are solid — such, for instance, as uniform rectilinear 

 motions parallel to agiven plane and varying as some function of 

 the distance from the plane, or uniform motions of revolution of 

 infinitely thin cylindrical shells with velocities varying as some 

 function of the distance from a common axis, or steady spiral 

 motions composed of these two kinds. So far as the motion 

 partakes of that which pertains to a solid, it must be determin- 

 able from the data of the problem by separate treatment, and 

 has to be eliminated. For the remaining motion udx + vdy + wdz 

 is integrable of itself, because this analytical circumstance spe- 

 cially indicates that the motion is such as pertains to a fluid. 

 Hence the reasoning (in the Phil. Mag. for March 1851, 

 p. 232, and June 1873, p. 436, and in the ' Principles of 

 Applied Calculation,' p. 186) from which I have inferred 

 that the motion is rectilinear when udx + vdy + ivdz is an exact 

 differential, and have thence attempted to get rid of the diffi- 

 culty resulting from the before-mentioned reductio ad absur- 

 durrij cannot be maintained. I have recently ascertained that 

 the origin of the difficulty admits of being simply stated as 

 follows. In the reasoning by which the rate of propagation in 

 a fluid defined by the equation p = a 2 p can be shown to be 

 such as to lead to the above-mentioned absurdity for a parti- 

 cular form of the arbitrary function, it is assumed that the 

 lines of motion are all parallel to a fixed plane, or that the 

 surfaces of displacement are planes. (See Phil. Mag. for Jan. 

 1851, example I., p. 34, and i Principles &c.' example L, 

 p. 193.) Now in that case, as is evident, the motion is not 

 such as is peculiar to a fluid, but of the kind for which, as 

 above said, udx + vdy + wdz is integrable by a factor, whereas 

 in the reasoning that differential expression is assumed to be 

 integrable of itself. This contradiction accounts for the re- 

 ductio ad absurdum. 



In order to clear up the difficulty, I have supposed that rec- 



