126 AN ESSAY ON THE ROOTS OF INTEGERS, 



as'. Then by Art. m'.) Since 6,747 ±z 15 ar <p° -f 12 a <p b -f 3 b 2 , so 

 387 + 6,747 = 7,134 = (6 a p b -f 3 b°) -j- (15 a 2 <p 2 -J- 12 a <p b -f 3 ¥) — 

 15 a 2 <p 2 -f 18 a<p b -f 6 Z> 2 , and is the upper number in the Rank of the 

 Square. 



y'. Then 7,134 X 3 = 21,402 = (15 a 8 <z> 2 -f 18 a <p b -f 6 & 2 ) 

 X b = 15 a 2 p 3 b + 18 a p b*~ + 6 b 3 , and is the product written in the 

 Rank of the Cube. 



z'. Then by Art. p'.) Since 199,343 = 20 « 3 f -f 30 ar p 2 b -f- 13 a <p b z 

 -f 4 b 3 , so 21,402 -f 199,348 == 220,750 - (15 a 2 f b -f 18 a p b 2 -f 6 Z> 3 ) 

 -|- (20 a 3 p 3 -f 30 a" f b -f 18 a p b 2 -j- 4 b 3 ) = 20 « 3 <p 3 -f- 45 a 2 p 2 b -{- 36 

 ^ <p b~ -f 10 Z> 3 , and is the upper number in the Rank of the Cube. 



u'. Then 220,750 X3 = 662,250 = (20 a 3 p 3 + 45 a" p 2 b -f 36 a p b 2 

 -f 10 P) X b — 20 a 3 p 3 b + 45 a" p 2 b 2 -f 36 a p b 3 -f 10 b\ and is the 

 product written in the Rank of the Biquadrate. 



/3'. Then by Art r') Since 3,535,365 = 15 a 4 f -f 40 a 3 <p 3 b -f 45 a 2 

 p 2 ¥ -f 24 a p b 3 -f 5 £*, so 662,250 + 3,535,365 = 4,197,615 = (20 a 3 p 3 b 

 -f 45 a 2 p 2 b" -f 36 a p b 3 + 10 5 4 ) -f (15 a 4 f 1 -f 40 a 3 p 3 b -f 45 a 2 p 2 b* -f 

 24 a p b 3 -f 5 Z> 4 ) = 15 « 4 p 4 -f 60 a 3 p 3 6 -f 90 « 2 p 2 b 2 + 60 a p b 3 -f- 15 5* 

 — 15 ( a 4 £ 4 -f 4 a 3 p 3 b + 6 a 2 p 2 b 2 + 4 a p b 3 + V) = 15 (a p -f & )\ and 

 since a p -f- & is = p by Par. 21, so 15 (a p -|- ^) 4 = 15 p 4 , and is the sum 

 written in the Rank of the Biquadrate. 



/. By the transference of 4,197,615, its units are put under the 5th 

 place of the third period, and hence 15 p* thus transferred, is the upper 

 number in the Rank of the Biquadrate. 



