110 



ProceedinfjS of the Royal Irish Academy. 



combined with Idx + nidn + ndz = 0. Let r represent either of the distances 

 from P of the points Q, Q' where the normal meets the consecutive normal. 

 Then liy expressing the condition that the normal at .r, ,y, s may intersect the 

 normal at a: + dx, y + dy, z + dz, at the point 5, t], 4, where § = .r + rl, 

 ))=?/+ rm, 2 = 2 + nil we find, since d^ = drj = dZ, = dr = 0, 



dx + rdl = 0, dy + rdm = 0, dz + rdn = 0, 



dX, djii, dn being total variations. 



and therefore elimiaate d.x., dy, dz the quadratic for - is 



0.1 + 



ii &2 + - 63 



T 



Ci 



C3 + 



= 0, 



where 



d/ dl , dm ^ 



«,= -=-, (iz = -r, bi^ rj- , etc. 

 dx dij do: 



the absolute term s'anishing, since I- + vv + ?!= = 1. The points Q, Q' 

 corresponding to the point P are then determined from the equations 

 g = .r + rl, etc. It is easy to see that as P moves along the Il-family 

 Idx + mdy + ndz = 0, Q and Q' move along some one of two different groups of 

 n-families (corresponding to the two sheets of a siu'face of centres) ; for |, ij, Z, 

 satisfy differential equations of the form 



pdX + qdt] + rdZ = 0, 2^V7S + q'dT!) + r'dZ, = 0. 



2^, ?. ''. p'> ?.' '■/ ai'G one-valued functions of x, 1/, s, but many-valued functions 

 of t,, »), 2. Thus the original ri-family is associated with a finite number of 

 other n-families, the differential geometry of which might yield some 

 interesting results. 



(j) If we use our special axes, it will be found that the quadratic for - may 



be written 



1 i /I 1 \ 1 , r- n 



----+-+ + i/==0. 



In fact the values of r, it is obvious, are the radii of curvature for the 

 directions of zero torsion [See (c).] The sum of the reciprocals of the values 

 of r is always equal to the sum of the principal curvature, and, if •/ = 0, it 



