1854.] 417 1^^^*' 



The intormediate alternate terms in the series (not distinguished by 

 the sign A) are only approximately R. A. triangles, having the 

 character which may be inferred from the following expression. 

 B' + P^r^H-'+l. 



The preceding table, derived from the initials " 1, 1," presents a 

 series of approximate common fractions of the square root of 2, a 

 quantity that can only be approximately expressed in limited terms. 

 As expressions of the value of the root of 2, the successive fi'actions 

 present discrepancies which have peculiar relations to y/l or 1. Other 

 series similarly derived from other initials, will, in a similar manner, 

 be approximate fractions of \/2, but each series will have its charac- 

 teristic discrepancy, which is related to the series as y/1 or 1 is to the 

 series just considered. This discrepancy will appear in any series as 

 the quantity D in the expression B — P=+D, which expression is 

 the characteristic of all the R. A. triangles of a series. 



The annexed series exhibits proportions simi- 

 lar to those of the preceding. The characteris- 

 tics of the triangles in the alternate terms (A) 

 in the form — n-j is B — P=:+ V 7. The approxi- 

 mate R. A. triangles (not designated by the 

 sign A) maybe characterized by the expression 

 B2+P2=H«+v/7. Thesuccessive fractions ?J- 



: \/2 approximately 



X y 



1— A— 2 



5 3 



11— A— 8 



27 19 



65— A— 46 



157 HI 



379— A— 268 



regarded as approximations to x/2, present dis- 

 crepancies which are related to v/7 or 7 as the 

 discrepancies of the preceding table are related 

 to v/l or 1. 



The triangles evolved by expanding the successive values of x and 

 y, as in the formula embracing those terms, have the characteristics 

 B— P=±7. 



This table having been derived from the initials " 1, 2," by addi- 

 tions, may be continued backwards by subtractions, which will de- 

 velop a series of terms, among which negative quantities will appear. 

 The first pair of terms (x and y) in which a negative quantity appears, 

 may be regarded as the initials of a new series, correlative to that 

 from which it is derived. The initials being found may both be re- 

 garded as positive, and the series extended by additions, as in the 

 preceding instances. 



X— y^J'- J—y>=^^", 5,— y'=y,„ &c. x,=+3. y'=— 1, 



the initials of the series. 



VOL. IX. — 3d 



