of determining Stream Lines and Equipotentials. 245 
Case (a). The guiding lines are everywhere orthogonal to a 
family of surfaces. — Let these be the surfaces over each of 
which 7 = F 3 ( t r, y, z) is constant. 
Then choosing a particular surface, say 7 = 7 , we wish to 
draw thereon a chequerwork of orthogonal lines, and we wish 
this chequerwork, by the motion of each point of it along the 
guiding line at that point, to sweep out two families of surfaces 
in space, in such a way that one family may be equipotentials 
and the other stream- surfaces. This requires that these two 
families, which we may denote by 
* = ¥ Y {x, y, z) = const., /3 = F 3 (>, ?/, z) = const,, 
should be everywhere orthogonal. Therefore the surfaces 
a. /3, 7 are mutual orthogonal, and consequently the surfaces 7 
must satisfy the condition necessary in any member of a triply 
orthogonal system (Salmon, ' Geoinetrvof Three Dimensions,' 
4th ed. §§ 476 to 486). 
But more than this. For we wish to be unrestricted as to 
the direction of the orthogonal traces of a and /3 drawn upon 
the surfaces 7. Therefore, since three mutually orthogonal 
surfaces necessarily intersect in their lines of curvature 
(loc. cit. § 304), it follows that at every point of the surfaces 7 
there are lines of curvature in every direction. The only 
form which possesses this property is the sphere or its limit 
the plane. Therefore the surfaces 7 are either spheres or 
planes. This is necessary. We have not proved that it is 
sufficient. As frequent reference will be made to the theorems 
proved in Lame's Lecons sur les coordonnees curvilignes, it will 
be convenient to employ expressions such as (Lame, § xi. 15) 
to indicate equation 15 of § xi. of this treatise. The relation 
of our notation to Lame's is that his p p 1 p. 2 are replaced by 
a /3 7 and that a ft 7 are used as subscripts respectively 
instead of absence of subscript, 1 and 2. Otherwise the 
notations are the same. 
In particular, if F is any function of position, we will 
denote by H F the quantity 
V(SHINf)" 
which is the reciprocal of the space-rate of F along the normal 
to the surface F = constant. 
Consider the lamina bounded by the two spheres 7 and 
ry + 87. The thickness of the lamina is K Y By. If a = V = the 
