The Rev. J. Challis on the Rectilinear Motion of Fluids. 101 



in excess and the other in defect, thejirst reciprocal equation 

 suffices to show their inequality. These points are, evidently, 

 the reciprocals of the integers above unity, that is' 



•5, -33, '25, -2, -167, '143, &c. 



It would at first sight appear that the least favourable case 

 for the separation of the roots is that in which they differ 

 least from unity, since this gives its maximum value to the 



denominator of the fraction j- ; but it must be remarked 



ao 



that this hypothesis involves the consequence that the corre- 

 sponding pair of roots in the reduced reciprocal equation will 

 have their smallest value, and therefore be in the most favour- 

 able state for separation. 



In order then to obtain some general insight into the extent 

 of separation effected by each reciprocal transformation, we 

 may take, as that of slowest divergence, the case in which the 

 roots occupy a point midway between two of the above numbers, 

 and are as near to *5 as this condition will allow. Assuming, 



therefore, the roots to be rather greater than *41, we have — -. r 



ao 



sa 6 ; and taking this number as the factor by means of which 

 may be determined the divergence of two roots having, ori- 

 ginally, their difference less than unity, it follows that the 

 number of figures in its wth power will, not inaptly, represent 

 the smallest number of figures which can be identical in the 

 roots which become separable in the wth reciprocal equation. 



From the preceding investigation we obtain a correct idea 

 of the extreme rarity of the cases in which the impossibility, 

 or the near equality of two or more doubtful roots can fail to be 

 made manifest by means of this simple method of reciprocals ; 

 and the improbability of the occurrence of these cases affords 

 the strongest evidence of the general utility of the method. 



Royal Military Academy, June 13, 1842. 



XVII. On the Analytical Condition of the Rectilinear Motion 

 of Fluids. By the Rev. J. Challis, M.A., F.R.A.S., Plu- 

 mian Professor of Astronomy and Experimental Philosophy 

 in the University of Cambridge* . 



THE mathematical reasoning which I gave in the April 

 Number of this Journal (S. 3. vol. xx. p. 281) respecting 

 a new equation in hydrodynamics, led me by indirect conside- 

 rations to the conclusion, that when ud x + v dy + wdz is a 



* Communicated by the Author. 



