we have 



232 Prof. Challis on tlie Principles of Hydrodynamics. 



cu'cumstauce, if it admits of interpretation M^th regard to the 

 motion independently of the consideration of particular instances 

 of motion, must refer to the mode of action of the parts of the 

 fluid on each other. I proceed, therefore, to use equation (3.) 

 in the solution of the foUomng problem. 



Proposition YII. Assuming \ to be a function of -^ and t, to 

 determine whether this supposition corresponds to any general 

 circimistance of the motion. 



Let -J- represent the ratio of the increment dato tempore of 



the function -^/r to the con-esponding increment ds of a line di'awn 

 constantly in the direction of the motion of the particles through 

 which it passes, and terminating at the point xyz. Also, let 

 Y = the velocity at that point at the given time. Then, since 

 generallv, 



d-^ d-^lr das d-yfr dy dsjr dz 

 ds dx ds dy ds dz ds 

 __ dyjf u d-^lr v </i/r w 



~"^"v "^ 'd^"\'^~d^'Y 



_ u^ + v^ + w^ _ V 



~ XV ~ X" 



Hence v— > *^^ 



ds' 

 But the general equation (3.), \iz. 



becomes by multiplying by X, 



Hence, substituting the foregoing value of V, 



f^-^ +%(')=« («■) 



As \ is by hypothesis a function of yjr and t, and as this equa- 

 tion is consequently a partial chfferential equation of the fii'st 

 order between yfr, s and t, it follows that •x/r is an arbitrary func- 

 tion of s and t. Hence by the definition of the function i/r, the 

 general equation of the surfaces cutting at right angles the direc- 

 tions of motion is 



t(*, 0=0. 



