107-108] 



ORTHOGONAL COORDINATES. 



157 



- 



w 



T 



(2), 



the direction-cosines of the normals to the three surfaces which 

 pass through (#, y, z) will be 



respectively. It easily follows that the lengths of linear elements 

 drawn in the directions of these normals will be 



respectively. 



Hence if <f> be the velocity-potential of any fluid motion, the 

 total flux into the rectangular space included between the six 

 surfaces X + \ SX, p J S/A, v \ &v will be 



7 



dv 



d ( d$ ^ %v 



T7 ( "1 ^V T" ' ~T 



d\.\ dX h 2 h 3 



, d 



-j- \h 2 y- . T- . -j- SyLt + -T- 



dv 



It appears from Art. 42 (3) that the same flux is expressed by 

 V 2 multiplied by the volume of the space, i.e. by S A //,&>//*! /* 2 /? 3 . 

 Hence 



d / A 2 \ d 



Equating this to zero, we obtain the general equation of continuity 

 in orthogonal coordinates, of which particular cases have already 

 been investigated in Arts. 84, 100, 104. 



* The above method was given in a paper by W. Thomson, " On the Equations 

 of Motion of Heat referred to Curvilinear Coordinates," Camb. Math. Journ., t. iv. 

 (1843) ; Math, and Phys. Papers, t. i., p. 25. Reference may also be made to 

 Jacobi, "Ueber eine particulare Losung der partiellen Differentialgleichung ...... ," 



Crelle, t. xxxvi, (1847), Gesammelte Werke, Berlin, 1881..., t. ii., p. 198. 



The transformation of v 2 to general orthogonal coordinates was first effected 

 by Lam6, " Sur les lois de 1'equilibre du fluide ethe"reY' Journ. de VEcole Polyt., 

 t. xiv., (1834). See also Legons sur les Coordonnees Curvilignes, Paris, 1859, p. 22. 



