130 AN ESSAY ON THE ROOTS OF INTEGERS, 



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

 added together, since by Art. y') tn at upper number — 15 p 4 so by Lem. 7) 

 the sum — 



20 f <p 3 X that Digit -f 15 p* <p s X that Digit | 2 -f 6p<p X that Digit | s -f 

 that Digit] 4 -f 15 jo 4 X p* = 15 jo 4 f -f 20 f <p 3 X that Digit -f 15/ <p* X 

 that Digit | " -f 6p <p X that Digit | 3 -f- that Digit | *. 



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



come 15 j»V l X that Digit -\-20p 3 p 3 X that Digit | 2 -f I5p 2 f X that Digit j 3 -f 



6p<p X thatDigit| 4 -f that Digit] 5 . Then if this product be written inthe 

 Rank of the Quadratus Cubi, opposite the third period, since by the 

 transference of Art. u', the units of the upper number in the Rank of the 

 Quadratus Cubi are put under the 6th place of the third period, so they 

 are also put under the 6th place of this product. 



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

 added together, since by Art. u', that upper number — 6 p 5 so by Lem. 7) 

 the sum = 15 p 4 p 4 X that Digit -f- 20 p 3 <p 3 X thatDigit| 2 -f I5p°-<tf x 

 that Digit | 3 -f 6p$ X that Digit | 4 -f- that Digit | s -f 6p 5 X tf = 6 p 5 £ s -f- 

 15 p 4 $ 4 X that Digit -j- 20 f <p 3 X that Digit |* -f- 15p* f x that Digit J 3 

 -l-6p<px that Digit | 4 -j- that Digit [ s . 



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

 become Qp 5 f X that Digit + \bp 4 f 4 X that Digit | 2 -f 20p 3 <p 3 x that Digit j J 

 -f 15 p Q <p" X thatDigit| 4 -f 6p<p X that Digit ( 5 + that Digit | 6 , which is requir- 

 ed to be not greater than R' <p 6 -j- C by Art./.) Now this is evidently the 

 same as the expression of Par. 23.) 6 p s f 5 c -J- 15 p 4 " <p* c 2 -f- 20 p 3 <p x c 3 -f- 

 15 p" f 2 c 4 -j- 6 p <p c s -}- c n , having that Digit substituted for c. And since 

 6 p s <p 5 c -j- 15 p 4 <p 4 c 2 -f 20 p % ft c 3 -{- 15 p" (p- c 4 -f 6 p <p c 5 + c 6 , must also 

 be not greater than R' <p 6 -f- C, and since this c must be a Digit, so it is 



