ye 
211 
such as the square of the velocity of a planet, a quantity of the 
first order. In Newron’s theory, which accounts very exactly for 
the motions, @:r is found to be of the same order as the square 
of a velocity; this we take from Newron’s theory. In (13) A must 
then be a quantity, differing little from 1; we represent it by 
„ua 
Az=l 4E 
A 
In (14) B is a quantity of order 4. We represent it by 
B=Va:a 
and take 4 positive. The constants 4 and u then take the places of 
A and B. If we substitute these constants in (21), this equation 
becomes 
rie 1 1 L fdr? 1 
Jt ge WEAR 
r a ar \d¢ 7 
Teese RO en! 
a jk 
The constants 2 and w are moderately great. The formula passes 
into the corresponding one of Nrwron’s theory, if we put «—0. 
We then obtain 
BREE hen fn 
2 7 dp 7 — u . . . . n . (2 ’) 
The equation gives rise to an ellipse, if u is positive, to a para- 
bola if u == 0, to a hyperbola if u is negative. In Newron’s theory 
Au < a‘. In consequence of the introduction of the constants 2 and u 
the equations pass into 
t te,t@,=1. er HEL LL S= U (A*+-ua), zer, =uea' . (22a) 
We see from these that the roots 2,, #,, #, approach very nearly 
to 1,0,0. The quantity a (32° + ue) is positive. Because u << 44° the 
roots prove to be all real. 2, is somewhat smaller than 1, about 
«2; x, and x, are of the order of a; they are both positive if wis 
positive, else they have opposite signs; 2, becomes zero if u — 0. 
We will therefore put 
2 = 1 4 om, 
ds a (m+ 2), 
Co (en): 
Now a, -+2,-+2,=0 as it ought to be; if n <{m we have to 
deal with the quasi-elliptic motion, if m > m with the quasi-hyper- 
14* 
