526 Prof. Sylvester's Notes to the 



FgrrL' + ^M' + ^N', 



%±M*, %=%&, U 8 =J?B*, 



r 8 r 3 x l 



V — — J H* V— ^R* V„—^iP* 



r 3 rj *j 



we may easily establish the following propositions, which indeed 

 are almost self-evident: — 



(1) Each U and V is a rational fraction. 



(2) When i= oo, eachU=Rs, each V=R*. 



(3) For all finite values of i, R* is intermediate between the 



least and greatest U, and R 1 * between the least and greatest V. 



So in general if k is any prime number, we may form (&— 1) 

 cycles, each cycle containing k fractions possessing precisely 



analogous properties as regards representing approximately and 



i 

 limiting the successive powers of R*. By means of these for- 

 mulae [the theory of which might be extended to algebraic quan- 

 tities of every order (in Abel's sense of the word)], we obtain a 

 complete command over the integration of surd quantities in 

 general as they may appear in any physical problem, being 

 thereby enabled to represent the integrals, not merely arithmeti- 

 cally, but analytically (which is of much higher importance) by 

 logarithmic and circular functions to any degree of accuracy that 

 may be required, and with known assignable numerical limits of 

 error. 



Note B. 



This note relates to the concluding paragraph of the long 



note at page 313 in the October Number of the Magazine. 



2 

 I find that the ith inferior limit to F(c) — log j, when c differs 



indefinitely little from unity given by the method therein ex- 

 plained, is 



2*-l 

 . 2 2 v *=i C ° S 2k w /. 2*-l y 



v 1+ ( sm _-v 



and that the superior limit is 



