169 
'-M ' 
perpendicular to CD. In New- 
ton's proof of the proposition that 
if the law of attraction he that 
of the inverse square the force at 
P is zero, the surface is divided 
into an indefinitely great number 
of opposite elements by small 
cones having their vertices at P, 
fhe attractions of each of 
these pairs of elements are shown to balance each other. 
We stall first shew that if the attraction at P be zero, then 
It follows inversely that for at least one position, if not for 
all positions of the cone MPm, besides the position APB, 
the attractions of the opposite elements balance each other 
and we shall thence prove that the law of attraction must 
u6 that of the inverse square. 
Let us supple the cone with vertex at P to move round 
rm the position where AB is its axis to any other position 
Pm. At AB the attractions of the opposite sections on P 
are equal whatever the law of the force. As the cone 
leaves AB let us suppose the resultant attraction of the two 
opposite elements to be no longer zero, but to act, say, towards 
the centre side of APB. Then it will either continue 
owards that side as the cone moves all the way round 
lom B^ to BPA or it will vanish at some position, and 
then act in the opposite direction. In the first ease we 
s ould have a number of forces all acting from P towards 
he same side of APB, wiiose resultant is zero. Then each 
separate force must be zero. In the second case the resultant 
attraction of the opposite sections vanishes somewhere 
etween AP and EP. Then for at least one position of the 
cone, besides the position APB, the resultant attraction of 
the opposite sections vanishes. 
Since this is true for any position of P, we can show that 
the law of the force must be that of the inverse square. 
