Rogers — Orthogonal Trajectories of a Congruence of Curves. 117 



The Divergence of the vector is 



— (IR) + ^r- (™^) + T- i.nB) = B[— + —\+ I -— + m —-— + n—-, 

 ax dy dz \pi p^J dx ay dz 



Vpi pz g 



Hence the Divergence of a vector is eqiuxl to twice the mean curvature of 

 the Il-family multiplied hy the magnit^ide of the vector, diminished by the 

 magnitude of the component, along the normal to this family, of the vector-gradient 

 of B. It will be noticed that the Curl is independent of the mean curvature, 

 and the Divergence is independent of the mean torsion. 



Irrotational motion, La'place's equation, etc. 



The following results may be of interest: — 



(a) The conditions for pure strain, irrotational motion, etc., viz., u = v=w = 0, 

 may now be expressed by saying that the mean torsion is zero ; the direction 

 of the gradient lies in the osculating plane of the line of flow or displacement 



<}> = 90°), and makes with this line an angle xL whose sine is + -V. Hence in 



P 

 this ease g is never greater than p. 



(h) We can now give a geometrical meaniiig to Laplace's equation V' F = 0. 

 Let X = ^— , etc., and the equation reduces to 



|o'' \pi pj g- 



since (b = 90°, cos = — , and sin li = ± cos 0. 

 P 

 (c) In general 



\Pl p2 9p J 



(d) A circuital vector is geometrically defined by 



n 1 



cos li = ff — + — 

 Vpi p, 



hence for such a vector ^r- must be greater than the mean curvature. 



2g 



