384 Scientific Proceedings, Royal Dublin Society. 
I may remark that these four circles treated in this way only 
produce two cones, as /4,=0 and /,=0 produce the same cone, so 
likewise do 7, and i. 
I shall now make a few remarks on a specially restricted system 
of tangential coordinates. Considering A, m, v as the parameters 
of the variable plane 
Av + py t+ve-1=0, 
it is required to find the relation between X, pu, v, which subsists 
when the plane passes through the pole of 
Az+ By + Cz+D=0 
with respect to the sphere 
et+yt+2—-1=0. 
Now, if this point be w’y’s’, we must have 
Av’ + py’ + vs’ -1=0; 
also, we must have 
yet ei ae ae ee 
Ti D’ Ys D ae. De 
therefore AX + Bu+ Cv+ D=0 
is the required condition, and it is called, in tangential coordinates, 
the equation of the pole of 
Azv+ By + Cz+D=0 
with respect to the sphere 
e+yt+2?-1=0. 
Similarly, 
an? + MhAp + bu? + 2gdv + Quv + ev? 
+20 + 2m + 2nv+d=0 
is the tangential equation of the reciprocal of 
aa + 2hay + by? + 2gua+ 28fyz + ce 
+ 2le + 2my + 2nze+d=0. 
Now, since the wave surface is known to be the envelope of the 
plane 
AL + Wy + VB =2, 
NG 74 vy 
2 ace = at yo 
va vi Ye 
and V+ w+’ =1, 
where 
ie 
