124 AN ESSAY ON THE ROOTS OF INTEGERS, 



£> 3 P 4 15 a 1 p" b* -f 6 a p b s -f Z> 6 ) and is the second Remainder, which 

 therefore agrees with the second Remainder found by the European 

 method in Par. 34, Art. d', and is therefore = R' (Par. 21). 



/. Then as in Par. 34 Art. ef) 18,535,911,758,593 expounds W p 6 

 4 C, and hence the second Resolvend of the European and Arabian 

 methods agree. 



"6 J 



V. Then by Art. g). Since 123 — 6 a p 4 5, so 3 -f- 123 = 126 — b 

 4 (6 «. <p 4 b) — 6 a p -f- 2 5, and is the upper number in the Rank of the 

 Latus. 



V. Then 126 X 3 - 378 = (6 a p 4 2 J) X b - 6 a p b 4 2 Zr, and is 

 the product written in the Rank of the Square. 



mi'. By Art. r). Since 6,369 = 15 or p z 4 6 ap 5 -j- 6 J , so 378 -f 6,369 

 == 6,747 - (6apb + 2**) 4 (15a 3 ^. -f 6 a p 5 + ft*) = 15« 2 <?r -J- 12ap/; 

 4 5 2 , and is the upper number in the Rank of the Square. 



ri. Then 6,747 X 3 — 20,241 = (15 a 2 p" 4 12 a p b 4 3 b 2 ) X b = 

 15 a? p" b 4 12 a p ¥ 4 3 b 3 , and is the product written in the Rank of the 

 Cube. 



p'. By Art. p). Since 179,107 = 20 a 3 p* 4 15 a 2 p- b 4 6 a p ¥ 4 5 3 , 

 so 20,241 + 179,107 = 199,348= (15 a? p" b 4 12 a p b" 4 3 b 3 ) 4 (20 a 3 <p 3 

 4 15 a? p- b 4 6 a p 6= 4 b 3 ) — 20 a 3 p 3 4 30 a 2 p 2 b 4 18 a p b~ 4 4 6 3 , 

 and is the upper number in the Rank of the Cube. 



q'. Then 199,348 X 3 = 598,044 — (20 a* p* + 30 a" p* b + 18apb 2 

 4 4 6 3 ) X b = 20 a 3 p 3 5 4 30 a* f & 2 + 18 a (dJ 3 -f 4 6 4 , and is the 

 product written in the Rank of the Biquadrate. 



