Lea.] ^2g [September. 



5-A—i 



13 9 



.SI— A— '-^'-^ 



75 53 



181— A— 128 



= v/2 approximately' -^^^ annexed series is the correlative of the last 

 I preceding, and is in all respects similar, except 



3 I I that it is derived from different initials. Either 



series being given, the other may be found in 

 the manner suggested. 



In any case when a series has a correlative, it 

 may be found in the same manner. In any in- 

 stance in which D in the expression B — P= 

 +D, is a prime number, or a multiple of a prime 

 number, and the triangle to which the expres- 

 sion refers, is a Prime Right-Angled Triangle, 

 as previously defined, an analysis of the triangle by the formula em- 

 bracing the terms x and y, will give values for x and y, which may 

 be extended into a series, which series has its correlative, as in the 

 preceding instance. If D is found to embrace several factors which 

 are prime numbers, each one of those prime factors may be found to 

 give rise to two series of values for x and y, which will be cor- 

 relative to each other, so that there will be twice as many series of 

 values of x and y as there are prime factors in D, and accordingly 

 twice as many series of R. A. Triangles in which B — Pz=±D as there 

 are prime factors in D. 



If D is the square of a prime number, there will be three series of 

 values for x and y, two of which will be correlative to each other, the 

 initials of the third being x=y=^D. 



Other generalizations might be suggested co-ordinate with these, 

 which, however, are yet incomplete, and are reserved for further con- 

 sideration. 



If any prime R. A. Triangle be resolved into the fractional form 



—---(=-) and a series of fractions be derived therefrom by additions 

 or subtractions, as in the preceding illustrations, the alternate corres- 

 ponding values of x and y in the series will embrace a Prime R. A. 

 Triangle in the form — |~(=-) , and all such triangles in the series will 

 have the same value for D in the expression B — P=+D. If the 

 series of values of x and y thus tabulated be expanded into a series 

 of triangles (by means of the formula embracing x and y) the tri- 

 angles thus evolved will be characterized by the expression B — P= 

 ±D'^, (D referring to its value in the former instance.) 



Any whole numbers whatever, when used as the initials of a 

 series under x and y, as in the preceding illustrations, will develop a 

 series of numerators x and denominators y, of common fractions ap- 



