118 AN ESSAY ON THE ROOTS OF INTEGERS, 



w. Then 6 X 2 — 12 = 3 a X a — 3 or, and is the product written 

 in the Rank of the Square. 



x. By Art. m.) since 12 — 3 a? so 12 -f- 12 — 24 — 3 or -(- 3 a c - 6 a 2 , 

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



y. Then 24 X 2 r= 48 — 6 «~ X a = 6 «% and is the product written 

 in the Rank of the Cube. 



z. By Art. p.) since 32 — 4 « 3 so 48 -f 32 — 80 r= 6 a 3 -f 4 a 3 — 10 a\ 

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



a. Then 80 X 2 = 160 == 10 a 3 X a — 10 a*, and is the product writ- 

 ten in the Rank of the Biquadrate. 



|3. By Art. r.) since 80 = 5 a 4 so 100 -f 80 — 240 — 10 a 4 -J- 5 « 4 — 

 15 a*, and is the sum written in the Rank of the Biquadrate. 



y. By the transference of 240, its units are put under the 5th place 

 of the second period, and hence 15 « 4 thus transferred, is the upper number 



in Rank of the Biquadrate. 



h. Then by Art. v.) since 0=3 a so 2 -f6 = 8 = «-f3a = 4«, 

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



s. Then 8 x 2 — 16 — 4 a X a == 4 a*, and is the product written in 

 the Rank of the Square. 



£. By Art. x.) since 24 — 6 « c so 16 -f 24 = 40 — 6 a" -f 4 « 3 — 10 a 3 ", 

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



7i. Then 40 x 2 = 80 = 10 a 1 X a — 10 « 3 , and is the product writ- 

 ten in the Rank of the Cube. 



