8 Prof. A. Gray: Notes on Hydrodynamics. 



9. If in (14) we take the step ds f parallel to the three 

 axes Ox, Oy, Oz in succession, and denote q' for these direc- 

 tions by u, v, w, we obtain the three equations of motion 



-+2a, sz9 sm^ = -^ 



g+2^,sin^=-^ 



^— + 2G> s ^sin6^=- 

 Ot 



(18) 



It is to be noticed that these three axes may be inclined at 

 any angles, so that (18) are equations of motion for a system 

 of any three axes. 



10. If we assume that the axes are rectangular, and write, 

 according to the usual notation, 



2£= 







2 V . 



"da 



B<^ 



9^= 



2f 





5y' 



(19) 



axes 

 same 



and regard 2f, 2t), 2£ as vectors associated with the 

 Ox, Oy, Oz respectively, the vector associated in the 

 way with any axis the direction cosines of which are I, m, n 

 is 



2{Zf + mr}-\-n£). 



If now I, m, n be the direction-cosines of the normal to the 

 plane ds, ds', drawn outwards from the diagram (fig. 2), this 

 vector is a) ss ' . Thus we can write (14) in the form 



^+2q(lg + m V +n£)sm0=--l^ r . . (20) 



(jt Q S 



This form is much less compact than (14), but from it we 

 obtain at once the usual form of equations (18) for the case of 

 rectangular axes. Putting 1 = 0, we get q' = u, qn sin 0=—v, 

 qn sin = iv. Similar results are obtained by putting m = 0, 

 ii = 0, in succession. Thus (18) become for rectangular 

 axes 



-**+.**— g 





-2w£+2u£ = 



_ & 



9.'/ 



2ur) + 2vrj = — 



_ S% 



d* J 



(21) 



These equations are usually derived from the Eulerian equa- 

 tions of motion for the rectangular axes. The forms, however. 



