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MATHEMATICS: L. P. EI SEN HART 
The cartesian coordinates x, y, z, of N and x, y,z ot N are solutions of 
an equation of the form 
a~ +b (1) 
which we call the common point equation of N and N . 
Let M and M be corresponding points of N and N. With M as 
center describe a sphere whose radius is the distance from M to the 
origin. Let Si and ^2 be the sheets of the envelope of the spheres as M 
moves over N, and let mi and ^2 be the points of contact with Si and 
S2 of the sphere with center at M. The null spheres with centers at 
fjLi and meet the tangent plane of in a circle C. These circles 
from a cyclic system, that is they are orthogonal to 00 1 surfaces.^ 
If h and / are any pair of solutions of the system 
^ = {l-h)a, ^ = {h~l)b, (2) 
OV ou 
the functions x\ y\ z\ defined by the quadratures 
- ii^^ - i^^.^y' - ^y' =i^y = = (3) 
bu bv bv^ du bu bv bv' bu bu bv bv 
are the coordinates of a net parallel to N, and the functions x\ y' , z\ 
defined by 
bx' _jbx bx' _ ^ bx by' ^ j^^y _ -j^y . - — l (4) 
bu bu bv bv' bu bu bv bv' bu bu bv bv 
are the coordinates of a net parallel to N. Moreover, the nets 
and are applicable, and consequently the function 6^ = Sx'^ — Sx" 
is a solution of the point equation of and N'. By the quadratures 
^ = 1 ^ ^ 1 d^' 
bu h bu bv I bv 
we obtain a solution 6 of (1). 
The functions Xi, yi,Zi, and Xi, 3^1, Si, defined by equations of the form 
Of ~ - fC\ 
X^ — X — X • Xi — X — X vO J 
6' e' 
are the cartesian coordinates of two applicable nets Ni and N\, which 
are T transforms of N and Ni respectively; a net and a T transform 
