107-108] ORTHOGONAL COORDINATES. Io7 



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

 pass through (a?, y ; z) will be 



i7 ) &quot;27 fl/2 ^/ r \ / 



tt/Lt ttyL6 a/JL 



, c?^c , c?y , dz 

 dv dv dv 

 respectively. It easily follows that the lengths of linear elements 

 drawn in the directions of these normals will be 



respectively. 



Hence if &amp;lt;/&amp;gt; be the velocity-potential of any fluid motion, the 

 total flux into the rectangular space included between the six 

 surfaces X }\, /* 3/i, y y will be 



d ( d&amp;lt;j&amp;gt; v &v\ ^ , d /, c^6 81; S\\ j, d fj d&amp;lt;f&amp;gt; 



- - - X + -- A 2 -r^ . - . -r- 6/z + -- 



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



V 2 &amp;lt; multiplied by the volume of the space, i.e. by 



Hence 



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, &quot; On the Equations 

 of Motion of Heat referred to Curvilinear Coordinates,&quot; Camb. Math. Journ., t. iv. 

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

 Jacobi, &quot; Ueber eine particulare Losung der partiellen Differentialgleichung ...... ,&quot; 



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



The transformation of v&quot;0 t general orthogonal coordinates was first effected 

 by Lam6, &quot; Sur les lois de 1 equilibre du fluide eth6r^,&quot; Journ. de VEcole Polyt., 

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



