596 
normal plane in P coincides with the plane of osculation in P’ of 
the curve which is the locus of P’, when P describes its path. P’ 
lies therefore always in the normal plane of P, while the velocity 
and the acceleration of P’, which lie both in the plane of osculation 
of P’, hence in the normal plane of P, are perpendicular to tbe 
velocity of P. Hence 
(ee) Um, x + (y a y') ny + (z F. z') Un, 2 —= 9 | 
Umm Van! os Umy Va,y! a Um,z Vaz!’ = as Oi, 
Um, x Ja, z +} Um, y au hk Ym, z ee == ie 
where vo, must be determined out of (la), Jas) out of (3). 
If we write the first of these equations thus: 
(7) 
x' Om,z + y' Vin, y eeh Un, 2 == & Um, x + Y Om, y + vine 
and if we substitute in the second member the expressions for v,»,x, 
Om,y» Vmz from (15), we find 
a Vin, x + y' Um, y aa Z' on, 2 = Sa = ny = Sz ee (8a) 
We write the second of the equations (7): 
… da dy 
Um, x S+gz sant kl + “ae + Um, y nr — pz ae a Be 
ante 
+ Um,z C + py'—- 1e ie = 
and subtract from this - 
da! dy dz' 4: ’ dm, “ - dvn, ao F dvm,z ER 
Vn an aus Vin, y HE =< Un, z dt v di + y di z di = 
dg dn de 
md — = 3 
which appears from (8a) through differentiation. 
If we now keep in mind that: 
dum, x 
Im,2= 
r QUm,z — Ten, y 
it appears that 
! 1 ! ds dj nee ct we 
& Jin, x == YImy =e 2Jn, z= mi = zl bm ae =F SUm, x == Ym, y == GUm, z 
or 
2'In,x + Y'Jmzy + Im = Et + yt GE we +o yg EST eps (BB) 
where El) = — zn + 9S — ry 
If we finally write the third equation of (7) 
