TO THE COMPARISON OF SEVERAL MAGNITUDES. 



67 



If we compare the last three values of A, which are in this case 

 A 10 , A 9 , A 8 ; we observe that the order of the quotients must necessarily be, 

 A 10 = p 9 A 9 + q & A 8 + r 1 A 7 ; that is to say, the computation must take the 

 form 



1 



1 



1 



1 



1 



1 



1 















2 



















4 



2 









3 

 1 



11 



3 



1 



1 



2 



8 



9 



3 • 







3 



33 



106 



109 



142 



390 



3229 



30763 



99137 



a 





1 



11 



35 



36 



47 



129 



1068 



10175 



32790 



b 







1 



3 



3 



4 



11 



91 



867 



2794 



c 









1 



1 



1 



3 



25 



238 



767 



a 











1 



1 



2 



17 



162 



522 



e 













1 



2 

 1 



16 

 8 

 1 



153 



76 

 9 

 1 



493 



245 



29 



3 



1 



f 

 (J 

 h 

 i 

 k 



Hence, if the values of q were written above and between the contigu- 

 ous values of p, and similarly with those of r, as in the subjoined scheme, 

 the computation carried from left to right leads to the ultimate values of 

 A, B, C ; when carried from right to left it leads to those of A*, A^ , 

 and so on ; but in each case it gives the same aggregate group of numbers ; 

 with a difference merely in position ; and hence, whenever the numbers 







1 





1 





1 





1 





1 





1 





1 









2 





4 





























2 











3 





9 





8 





2 





1 





1 





3 





11 





3 



Po>Pi>Ps > Qo > 5i ' #2 , • • • , &c., are arranged symmetrically, the series 



of values A , A x , A 2 . . . .is identic with A x B x d read inversely. 



The continuation of the same process to the case of a greater number of 

 magnitudes is so obvious as to stand in no need of farther illustration. The 

 application of this process to problems involving the higher powers of numbers 

 may be expected, as the Brounckerian process has already done for squares, to 

 throw considerable light upon that difficult branch of the Theory of Numbers. 



VOL. XXVI. PART I. 



