601 
a surface O,, of the fourth order (6 7); in the same way a surface 
0; of the fourth order will be the locus of the points P’ the paths 
of which in the inverse motion have a contact of the fourth order 
with the spheres of curvature in P’. 
The former of these surfaces was determined by joining to (8) 
the equation (8d), which results from (8c) through differentiation, 
dx’ dy’ dz’ 
when for F ‚— the values are substituted following from 
de i eas 
Var! = Vay’ = tar = 9 
Now we do not require O'y, but the locus of the points P that 
are the centres of the spheres of curvature which have with the 
path of a point P/ of O', in the inverse motion a contact of the 
fourth order. 
The normal plane of the path of P’ has for equation: 
NO= (X—2’) oy + (Y—y’') vy + (Z—z2') wee ==) tard LO) 
the centre of the sphere of curvature is found from: 
aN© ad NO 
NO) ooh A ag ee (11) 
dt dt? 
the condition that this sphere has with the path of P’ a contact 
of the fourth order is expressed by 
a Nl) 
hie oe as ee Wie ce 
de dy dz' 
while Bl oe zp we determined from 
Var! == Vay! = Var = 0 
The locus in question is then found by eliminating a’, y’', z' from 
(11) and (12). 
In the same way however -— for the first member of (10) is 
identical with that of (8) — we have produced the surface O,,; the 
locus in question is therefore Os 
In the direct motion a surface On of the fourth order is the locus 
of the points fixed to T„ the paths of which have jive points im 
common with the spheres of curvature; in the same way in the inverse 
motion a surface O'y of the fourth order is the locus of the points 
fixed to Ty the paths of which have five points in common with the 
spheres of curvature; the points of Om and O'y correspond in the 
two cubical correspondences, so that e.g. in the direct motion a point 
