In a previous paper^{1} I worked out the general form of Newton’s Law of Gravitation and applied it to the special case of a planet orbiting the sun. In this case Newton’s Law was modified by the factor
1/(1  v²/c²) ^{�} 
For the case of an object moving directly toward another object rather than orbiting, the genetal equation reduces to Newton’s Law multiplied by the factor
(1  v²/c²) 
This is exactly of the same form as Lorentz’s modification of Coulomb’s Law.
Before applying the new factor, it is important to realize that the galaxy cannot be represented as a pofnt mass; rather it should be represented as a flat disk. The Newtonian actraetion of a flat disk for a point mass has been worked out before², but will be repeated here.
In Figure I let the radius of the disk be r and let its surface density be s . I aim to find the attraction of the disk for a polnt mass located at P on the perpendicular line passing through the center of the disk. Let 0 be the origin of a system of polar ccordinates p and q, and let z be the distance along the line to the attracted location P.
Since pdpq is the area of an element in polar coordinates, the mass of such an element is
dm = spdpdq 
(1)

The dietance of the element dm from P is
R = (p² + z²)^{½} 
(2)

and the attraction of the mass dm for the mass at P is
 G dm = Gspdpdq  
— ———— 
(3)

R² p² + z² 
and the component of the attraction along the exis is
 Gspdpdq . z Gspdpdq  
———— – =  ———— 
(4)

p² + z² R (p² + z²)^{3/2} 
The total intensity of attraction of the disk for the point P mass is
ò  n  ò  2p  pdpdq  
I = G s z  ————–  
o  o  (p² + z²)^{3/2} 
ò  r  pdp  
I = 2pGsz  ————–  
o  (p² + z²)^{3/2} 
[  z   z  ]  
I = 2pGs  ————  —— 
(5)


(z² + r²)^{½}  (z²)^{½} 
Assuming z positive,
[  z  ]  
I = 2pGs  ————  1 
(5¹)


(z² + r²)^{½} 
Now with the modifying factor included, the acceleration of the point masa toward the disk is
dv  [  z  ]  [  1  v²  ]  
— = 2pGs  ————  1  —  
dt  (z² + r²)^{½}  c² 
dv 2pGs  [  z  ]  [  c²  v²  ]  
— = ——  ————  1 
(6)


dt c²  (z² + r²)^{½} 
dv = dv dz = v dv  
— — — — 
(7)

dt dz dt dz 
The crucial deduccion in Larson’s gravitational theory is that the gravitational force of any mass extends outwatd only a finite amount the gravitational force does not extend out to “infinity”, as commonly assumed. At the gravitational limit of the galaxy, which will be denoted by d_{o}, the attracted velocity of a mass is zero. This velocity becomes larger to the degree that the mass is loeated closer to the galaxy. Let the velocity be v at distance z. Then, separating the variables in equation 6 and integrating between the limits, the result is
ò  ^{ v} 
vdv

ò  ^{z} 
^{ } 2pGs

zdz

  ò  ^{ z} 
2pGsdz


————— =

———–

————

————

(8)


_{o} 
c�  v�

d_{o} 
c²

(z² + r²)^{½}

d_{o} 
c²

The outcome of this result is that
v  [   4pGs  [(z² + r²)^{�}  (do² + r²)^{½ }+ d_{o  } z] 
½

]  
– =  1  e  ——––  ½ 
(9)


c 
c²

For our galaxy the constants in the equation are as follows:
G = 6.67 x 10^{11} N  m²/kg² 
c² = (3 x 10^{8})² m²/s² 
s = .2975 kg/m² 
r = 4.626 x 10^{20} m 
d_{o }= 2.177 x 10^{22} m 
With these values equation 9 becomes
v  [   2.771 x 10^{27}[(z� + 2,140 x 10 41)�  9.087 x 10^{13}  z]  ]  
– =  1  e  ½ 
(10)


c 
Speed in km/sec vs. distance ia kiloparsecs is plotted in the graph (Fig. 2). Great caution must be used in applying equation 10 to real masses:
 A globular cluster or a small galaxy associated with the Milky Way galaxy is not really a point mass; in fact, observation shows that the near side stars of such objects are attracted at the expense of the farside stars.
 Globular clusters are not falling directly toward the galactic center; rather they are orbiting.
 Small local galaxies are at a distance close to the gravitational limit of the galaxy — their veloclties aze difficult to measure and compare with theory.
Even so, the calculated velocities, neverthaless, agree in a very general vay with those observed for the local group of objects.
References:
1. Satz, R.W. REClPROCITY Vol. IV, No. 2, p. 25, July 1974,
2. MacMillan, W., The Theory of the Potential (New York: McGraw Hill Book Company, 1930),pp. 1516.