346 



Write 



Dr. T. Craig on Steady Motion in 



0=V<&>* + <£>| + <l>§; 



then we see from (9) that the stream-lines and vortex-lines 

 always lie on a surface the direction-cosines of a normal to 

 which are 



5i ^1 5? 



a' [XT n' 



We shall confine ourselves at present to the case where 

 ^ d® s 



dx 



%. e. 



® x dx + ^dy + <3> 3 dz 



is an exact differential d®. It is clear that, in order that this 

 quantity may be an exact differential, it is necessary only that 



"{(t-f)*+(§-S*+(i-§M 



or, briefly, 



Adx + Bdy + Cdz 



shall be such a quantity. The conditions which have to be 

 fulfilled on this supposition are well known to be 



dA dB ~ p 

 or, substituting for A, B, C their values, 



v 2 f=o, v 2 <?=o, vr=o. . . 



Since, now, 





(10) 



(ii) 



equations (5) become 



eft/ \dx dzP 



(12) 



