174 NEWTON S PRINCIPIA. 



reciprocals of the (nl) ih powers of the distances are in 

 arithmetical progression, the densities at those points will 

 be in geometrical progression. 



Two cases of the above are worthy of notice, when 

 n = 2 and when w = 0. In the former the force attracts 

 inversely as the square of the distance, and the density at 

 any point is given by 



+ ] 



p = D . g * * 



that is, if the distances be in harmonical progression, the 

 densities will be in geometrical progression. In the latter 

 case the force is constant and equal to p, and the density 

 is given by 



that is, if the distances decrease in arithmetical progression 

 the densities will decrease in geometrical progression. 

 These cases we might suppose to bear some analogy to the 

 state of our atmosphere, the former holding when the 

 changes of elevation are great, the latter when they are 

 small. 



There is one case, especially considered by Newton, in 

 which the preceding general formula fails, viz. when 

 n=l, for then log p in equation (3) appears to be always 

 infinite; but this is not really the case, for C is also 

 infinite and negative. The form of the integral has 

 changed, and by merely repeating the process, we get 



K log p = C p log 



x 



_ 

 .*. p = D . x * 



The preceding investigations are not, however, of any 

 very great practical utility. They are all founded on the 

 supposition that the compression varies as the density. 



