24:6 Mr. L, F. Richardson on a Freehand Graphic way 
potential, then the surfaces /3 = constant are surfaces of flow. 
Denoting in like manner the distances between two adjacent 
surfaces of these families by H a Sa, HpS/3, we see that H a £a 
and HpS/3 are the length and breadth of a chequer traced on 
the surface y. 
A tube of flow is bounded by the four surfaces /3, /3 + S/3, 
7, 7 + S7. And its cross section is therefore H^ . H y . 8/3 . 87. 
Now if the flux has no divergence, then along a tube of 
flow magnitude of flux multiplied by cross-section = constant. 
But the magnitude of flux is equal to the negative space-rate 
of the potential a along the line of flow, and this is —75-* 
Therefore along a line of flow — ^U — - • S/3 . By must be con- 
stant in order that the vector space-rate of the scalar a. — the 
Hamiltonian V« — shall have no divergence. Or equivalently 
the condition that 
V 2 a=0 is 
Now we may by freehand trial and amendment so arrange 
the orthogonal lines on the surface 70 that the above relation 
shall hold true on the surface 70; but we must further 
enquire what conditions the spheres 7 must satisfy in order 
that ^H — tj — ~ 1 =0 sna U De tme ^ or a ^ values of y when 
it is true for one y ; and this moreover when a and /S are 
otherwise undetermined. 
At this stage the fact that the surfaces y are spheres makes 
a remarkable simplification. For supposing for a moment 
that they did not possess this property and that rj and r? were 
their principal radii of curvature at any point, then by 
(Lame, § xxx. 24) 
1 _ A y "dJi a L = ^Z . S^/a 
*5 " K W *•? h ' W 
where h = m and similarly for fi and y. 
H 
Equating the two curvatures 
~dy a By 
, ~dh a "dhp 
ha .< — =h B 
