599 
dz! ¢ 
aie = (6 77 192 )- 
If however we differentiate (8a) this substitution gives us only 
(86); from (86) results in the same way (8c); ea (8c) we find 
ef HJ HJM |E ns AG E— + © :)| 
or 
‚78 ‚73 ' 
Boe Ieee a ee re 
where the expressions for the coefficients A,, EO a! 13 obs are rather 
extensive, but easily calculated. 
For the required locus we find accordingly : 
Um,x Um,y Um,z = Sx 
I nize Jy J mn, z =o — = gl? 
2 2) 2 = ; =— 0 
co. Jy sie 3 Poe a = §2 aa 
3 3 3 
Dee ee ued A, nn ze x | 
The points P of the system fixed to T,, the paths of which relative 
to Te have with the spheres of curvature in P a contact of the fourth 
order, form a surface of the fourth order. 
§ 8. It is clear that we shall find the same results when we 
consider the singularities of the inverse motion; at present we men- 
tion especially that a cubical correspondence will exist when we 
conjugate at a given moment a point P’ of fixed space to the centre 
P of the sphere of curvature of the path that P’ describes relative- 
ly to 7. We shall discuss the latter cubical correspondence 
more closely. 
The condition that P’(«’,y’,z’) shall lie in the normal plane of 
the path of P(a,y, 2) relative to Tf is expressed by the first of the 
equations (7): 
(w 
#) Ome + (y'—y) Umy + (2'—2) Omz = 0. 
Now 
(t'— 2) vm + (Yy'—y) Uy + (2 —2) Ume = (wr) (8 + gz—ry) + 
+ (y'—y) (4 + ra—pz) + (2'— 2) (0 + py—ye) = 
(#'—a) (§ + gz'—ry') + (Wy) (n + ra'—pz') + (2'— 2) (b+ py'—ge'), 
hence according to (5) 
(2'— 2) ome HWI) omy + (2'—2) ms = (el) 
i We 
+ (y—y) wy: srilez7e ) Une 
