1895.] of a Viscous Incompressible Fluid. 321 



is such that any one of the surfaces belonging to the family 

 (f) = const., may be considered as the locus of a singly infinite 

 series of the curves constituting the congruence. 



It is clear that there are an indefinite number of congruences 

 satisfying this condition. Another would be that given by the 

 differential equations 



dx 



8^' 



,^_,|)(^^,_,|)_(,^_,|)(,^_, 



"by 



Suppose that I, m, n and V, m, n refer to two distinct con- 

 gruences of this type, then 



doc dy oz 



and r ^ + m' ^ + n'^ = 0. 



ox oy oz 



Therefore 



d^ d(f) dcj) 



dx dy dz , 



— ? r- = ,, n = 7—! jT- = '^ say ; 



mn — on n nt — n i Im — i m 



from which it readily follows that 



d<f> + h {I' (mdz — ndy) + m' (ndx — Idz) 



+ n'{ldy-mdx)} = (20). 



Now mdz - ndy = k \^^ dz - ^-^^-^ dy^ 



VOL. VIII. FT. V. 



24 



