492 Prof. Helmholtz on Integrals expressing Vortex-motion, 



We will employ the symbol ~^~ in what follows only in the 



sense that -~ dt is the change of yjr during dt for that element of 



the fluid whose coordinates at the commencement of dt were x,y,z. 



If we eliminate p by differentiation from (1), and with the 



help of (2) introduce the new expressions, supposing (1 a) to be 



true for the forces X, Y, Z, we obtain the following equations: — 



^,_c : du du ydu 



St 



Srj 



dx 



dz' 



dv 



St"^dx' tV dy^ i dz i 



Tt~^Tx Jry] aYj Jr ^Jz' 



or, which amounts to the same, 



8f _ t d u dv y dw 



Srj _c.du 

 8t"^dy 



dv <,dw 



at 



dt 



du 

 Tz 



£/ ~~ V J_ + ^ ,7~ +Sj .• 





5" 



(3) 



(3«) 



If in a fluid element f, 77, £ are simultaneously equal to zero, 

 we have also 



Sf S77 8£ 



8/ & & 



= 0. 



Hence those elements of the fluid ivhich at any instant have no 

 rotation, remain during the whole motion without rotation. 



We can apply the method of the parallelogram of forces to 

 rotations. Since f, rj, f are the angular velocities about rect- 

 angular axes, the angular velocity about the instantaneous axis is 



and the direction-cosines of that axis are 



f % £ 



9 4 9 



If we now take in the direction of this instantaneous axis the 

 indefinitely small portion qe, its projections on the axes are ef, 

 67], and e£. While at x, y, z the components of the velocity are 

 u y v, iv, at the other end of qe they are 



