367 



multiplied by 2d, the co-efficient of the second term. There- 

 fore, the second ditTerences of the squares =rfX2rf. 



FXRST PIFF£R£NCES. 



2dq + d 



p = q + 2dq + d 

 q = i--{-2dr-^ d 

 r=s + 2ds + d 



2dr + d 

 Ids+d 



s=tScc. 

 The third differences of the cubes will be the second dif- 

 ferences of their first differences ; i. e. of the remainders, 

 after taking the highest members away. But substituting 3 

 for n, the second differences of the remainders —3d multi- 

 plied into the second differences of the scjuares, since the 



/A 



second differences of the one powers =0. Therefore the 

 third differences of the cubes =6^x2rfx3rf. 



FIRST DIFFERENCES. 



p=g + 3dq+3dq + d 



3 3 X 13 



q=r +3dr + 3dr + d 

 r=s +3ds +3ds + d 



3dq+3dq + d 



3dr+3dr + d 



ill 

 Sds+Sds+d 



s=:&c. 



In general, when the proposition is proved of all the 



powers whose Index is less than n, (as it has been proved of 



the second and third powers), in this manner also the case of 



tf> th 



the n powers will be deduced, viz. The n — 1 differences of 



the 



