40 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Our late colleague, General Alvord, in confutation of this not un- 
usual misconception, made a communication to the Society some 
two or three years ago (as those present doubtless remember) in 
which he showed that as gravitation was known to act equally on 
every particle of matter (7. e. proportionally to the mass) and as 
solid homogeneous spheres subtending any given conical angle from 
a center of reference possess volumes (or masses,—d being constant) 
directly proportional to the cubes of the conical altitudes or radii 
of distance, it follows—if gravity were a radial emanation—its 
effect must obey the law of inverse cubes of distance, contrary to the 
facts of observation. 
The fallacy here criticised springs evidently from the too common 
tendency to regard gravitation simply as a central force or as a 
single influence radial in direction, whereas it is always a duplex 
and reciprocal action; and however insignificant one of the terminal 
elements its presence and measure of distance cannot be neglected 
without completely nullifying all action. Thus m and m’ being 
two masses at any given distance apart, the action in the direction 
and through the distance m/’ m, is as real and positive as that in the 
direction and through the distance m m’. In other words, it would 
seem that the mutuality of the re-action necessarily involved with it 
the idea of reciprocity of the distance relation. Thus, adopting the 
suggestion of Mr. Bates, if we write the formula of the effect as 
(m +- d) X (m’ +d), we have this reciprocity distinctly brought 
out, and obtain at once the Newtonian formula. Thespeaker wished 
to learn from those more conversant than himself with mathemati- 
cal literature whether the suggested modification is new, and also 
whether any mathematical objection appears to its form. 
Mr. Hit remarked that one fault of the notation proposed ap- 
peared to be its want of generality, as it is evidently inapplicable to 
any other force having a higher or different exponent of the space 
function. 
Mr. Dooxtrr.e observed that, admitting the “reciprocity of the 
distance relation,” he yet failed to perceive how this function could 
appear in the formula as a product. Why should we write the dis- 
tance twice taken—as d multiplied by d rather than as d plus d? 
Further remarks were made by Messrs. Etuiorr, Bares, and 
ROBINSON. : 
