Lea.] 



416 



[September. 



angles thus evolved may be analyzed without the introduction or 

 suppression of factors). If in the above formula for prime R. A. 

 Triangles, x+y be substituted for N, and y for S, the following for- 

 mula will appear: H=x2+2 xy+2y-, P=x=+2xy, B=2xy+2y-. 

 This second formula will be chiefly referred to in the following sum- 

 mary : 



DECOMPOSED FRACTION OF THE SQUARE ROOT OF 2. 



""^ I S 5 J 2 2 9 7 I 6 9 "-•"• A - = — 



l/2=-l+j_ _ 



2 + l_ 



2 + 1 



2+1 



(A 



A 



A 



A 



B+P 



= \/2 approx. 



X y 

 i-A-i 



3 2 



7 -A— 5 



17 12 



41— A— 29 



99 70 



239— A— 169 



The decomposed fraction of v^2, when re- 

 solved into a series of numerators and de- 

 nominators of common fractions (-) will 

 present the values in the annexed table, 

 column X embracing numerators, and column 

 y the corresponding denominators. The 

 first two terras, or initials of the series, 

 being found, succeeding terms may be found 

 by additions, observing the following rela- 

 tions, x+y=y',y-fy'=x', x'-hy'=y", y'+y" 

 =x", &c., continuously; or, y+2y'=y", y'+ 

 2y"=y"', y"+2y"'=y"", &c. 



In this series it will be seen that each alternate fraction (A) em- 

 braces a triangle in the form — -— in each one of which triano-les is 



" H ~ 



a common characteristic, having the expression B — P=z+1- 



An analysis of the several triangles of this series, by means of the 

 formula embracing the terms x and y, will reproduce the series of 

 values of x and y respectively, as given in the table. This is the 

 only series which will reproduce its radical elements, for the reason 

 that ^^1=1; and the root of no other quantity than 1 is ecjual to 

 itself. If the several triangles be analyzed by the formula embracing 

 the terms N and S, N and S will successively reproduce the series of 

 denominators y, N being in advance of S. 



The general character of this and any similar series of triangles, 

 suggests the expression " —^—=^2 approximately." Other series of 



•triangles similarly derived from different initials will confirm this 

 suggestion. 



