376 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



Therefore by substitution, 



3J+-^+ *>(?♦?)"-• <5 >- 



I have obtained this equation in an entirely different manner in 

 the Transactions of the Cambridge Philosophical Society, Vol. V. Part u, 

 p. 196. 



5. The equation (3), by substituting -^- for u, -^ for v, and 



-^ for w, is transformed into another, which by integration gives, 

 ax 



Now since V = -¥- , -j- = t— ri- Hence -^ = f—rrds, tne m - 



tegration being performed along the line s. The above equation thus 

 becomes, 



<- «•+'/£*'+'?.?*«> (7) - 



Hence, by differentiating with respect to t, 



dP a 1 dp dQ rd'F ' „dV „, A . 



dt pdt dt J dt 2 dt 



and by differentiating with respect to s, 



dP a % d P dQ dV T _ dV 



rf,? ' pds ds dt ' ds 



Consequently by substituting in equation (5) we obtain, 



If now -3- be substituted for V, we have f—fjr ds — J . TL rf* = -7^-, 

 and the above equation becomes, 



