700 PHILOSOPHICAL TRANSACTIONS. [annO ]7u6. 



1.1.1 



\ '-i X : 1 i- - 



3l. 'I = 



,_ 5 ^-x = ^ + .-:i^ + 9-:^. + T3TF'^-- 



which two series are evidently 



C^ 9 -3 ^ 7. SI ^ 11.81^ ^ 15.81" • 



And Mr. Jones's other series, ? + ^ + ^^ + -^, &c. ( = l. J) is evi- 

 dentIy:=|x:l + ^^,+ 3i^ + ^., &c. = | X : 1 + ^ + -^ + 

 -, &c. We therefore now have 3l. 2 + l. -- equal to the sum of these 



7 . SI 

 3 series, 



9 3^ Ti.Sl ^ 11.81^ ^ 15.81*' 



2 111 



9 ^ ■ ' "^ sTsT + TTsT^ + TTsiT' ^'^• 



which sum is also equal to the hyperbolic logarithm of 10. 



4. The form to which Mr. Jones's series are now brought, is evidently the 



the same with the general form n X ■ 1 ^ 1 1 , &c. 



the value of which, while m and n are affirmative numbers, and x sufficiently 

 small, will be given by the series 



1 m" , n . 2« . .i'" 



a X : — A ^ &c * 



7n(l — r") m {m ^ n) . {\ — x^Y m (m + n) . (m + 2«) • (1 — >r"j* 



And this series, if we call the 1st, 2d, 3d, &c. terms oi" it a, b, c, &c. respec- 

 tively, and put _ „ = z, will be more concisely expressed thus ; 



1 ni\ . 2rtZB one , 4fl;D . i ■ i /- 



a X '■ — ; ; , r ■ — r-A r^ ; — r'y <*c. which form is 



well adapted to arithmetical calculation. 



Now, by comparing the 3 series at the end of the last article with the general 



series here given, we shall find that, in the first and last of these series, the 



value of 7n is J, and in the 2d it is 3. The value of n in the 1st and 2d series 



is 4, and in the 3d it is 2. The values of a are obviously 2 in the first series, 



and -I- in the 2d and 3d. But in each of them z, = — - — , is = — ^-^ = — , 

 ^ 1 — .r" 1 — y, 80 



These values of the letters being written for them in the 2d general form, we 

 have 3 new series, viz. 



* See Phil. Trans, for 1794, p. 218, where this matter is more fully explained,^— Orig. 



