( 628 ) 
a = RJi I cos n 1 t, y = — R t h V \ — g cos (2 n, t — tp ) and ^ 
5 Kl—£ cos y = A, 
we find for the equation of the osculating curves with g as parameter: | 
? (X* + n + s (xr - X 1 - x*) + ^x>-axrr+ x<) =|| 
where for the sake of a simplified notation is put: 
X for —r for 
R 0 ti R,h 
Thus the envelope has as equation 
4 (X* + Y') Q 1C- - 2 KX' Y + X‘j - (X Y - X' - X'f = 0. 5 
After reduction and division by X 2 (the P-axis is the locus of the 
nodes) it may be written: . 
(K -4 r s - 3 X* Y -f Yy = (X s -f 4 Y* — 1)’ (X* -f Y*% 1 
or if we solve K: 
K=-(Y± VX'+Y>) + (Y ± KX'+T*)*. 
Putting 
*r=t Kx*4-y* =—, 
^ u' 
K=-Tr + 
u 2 (1-£/)-*» = <>. 
Now this cubic equation has the same coefficients as (14), so il l 
also has the same roots. So the envelope is degenerated into the,: 
3 parabolae having as equations: 
17 = $, , .■#=£, ' , U= — k; 
which after reduction and reintroduction of .a and y take the form of: 
’ WJh’ + f =0 
S, parabola, 
K 
i + ^ = 
The parabolae are con focal and have 0 as focus. When K is 
positive the and the parabolae have their openings turned upwards, 
