120 AN ESSAY ON THE ROOTS OF INTEGERS, 



±z 6 a, so with the annexed Digit the whole figures will, by Lem. 6, become 

 — 6 a p -f that Digit. 



2. Then if that Digit be multiplied into these figures, the product 



will become 6 a p X that Digit -f that Digit. | z Then if this product be 

 written in the Rank of the Square opposite to the second period, then 

 since by the transference of Art. v, the units of the upper number in the 

 Rank of the Square, are put under the 3d place of the second period;, so 

 they are also put under the third place of this product. 



3. Then if this product and that upper number be in this situation 

 added together since by Art. v) that upper number — 15 a 3 , so by Lem. 7, 

 the sum = Gap x that Digit -f that Digit j 3, -f 15 a 2 X p 1, = 15 a" p z -f 

 6a<p X that Digit -(- thatDigit| J . 



4. Then if that Digit be multiplied into this sum, the product will 



become 15 a 3 p z X that Digit -f 6 a p X that Digit [* -f that Digit j 3 . Then 

 if this product be written in the Rank of the Cube, opposite to the second 

 period, then since by the transference of Art. r, the units of the upper 

 number in the Rank of the Cube, are put under the 4th place of the second 

 period, so they are also put under the 4th place of this product. 



5. Then if this product and that upper number be in this situation 

 added together since by Art. t, that upper number — 20 a 3 so by Lem. 7, the 



sum = 15 a % p" X that Digit -f a p X that Digit j 5 + that Digit ['■ -(- 20 a 3 



X p = 20 a 3 p l -f- 15 a" p" X that Digit + 6apX that Digit] 2 -f- that Digit| 3 

 6. Then if that Digit be multiplied into this sum, the product will be- 



come 20 a? p 3 X that Digit -}- 15 a" p"' X that Digit| -j- 6 a p x that Digit! , 



-f that Digit. | 4 Then if this product be written in the Rank of the 

 Biquadrate, opposite to the second Period, then since by the transference 



