128 AN ESSAY ON THE ROOTS OF INTEGERS, 



*.'). Then 135 X 3 - 405 = (6 a p -}- 5 h) X b n C a p h + 5 V\ and 

 is the product written in the Rank of the Square. 



ft/. Then by Art. £'.) Since 7,530 = 15 a" p" -f- 24 a p 6 + 10 b" so 

 405 -)- 7,530 = 7,935 — (6 a p b -f 5 b~) -f (15 a 2 p 2 -j- 24 a p 6 -f 10 6 2 ) — 

 15 a* p 2 -j- 30 a p b -f- 15 Zr — 15 (a 2 p 2 -}- 2 a p b -(- £» 2 ) =: 15 (a p -f £>) 2 , and 

 since a p -j- b is — pby Par. 21) so 15 (a p -J- ^) 2 = 15 j? 2 , and is the sum 

 written in the Rank of the Square. 



/. By the transference of 7,935, its units are put under the 3d place 

 of the third period, and hence 15//, thus transferred, is the upper number 

 in the Rank of the Square. 



f. Then by Art. »'.) Since 135 = 6 a p -f 5 b so 3 -f 135 = 138 — 



b-\-(6apJ^5b)=z6ap^-6b — 6(ap-{~b), and since a p -\- b is — p 

 by Par. 21) so 6 (a p -J- b) = 6p, and is the sum written in the Rank of 

 the Latus. 



•r'. By the transference of 138, its units are under the 2d place of the 

 third period, and hence 6 p, thus transferred, is the upper number in the 

 Rank of the Latus. 



V. Then if there be a Digit annexed to the right hand of the upper 

 number in the Rank of the Latus, since by Art. t 7 ) this upper number = Qp 

 so with the annexed Digit, the whole figures will, by Lem. 6) become — 

 6 p p -j- that Digit. 



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



will become 6p p X that Digit -f that Digiij 8 . Then if this product be writ- 



