444' Mr. Graves's Explanation of a remarl-able Paradox 



may be denoted by / — , since it may be similarly proved 



to be an e-log of — , and since it is the same indivi- 

 ^ a; 



dual function of — that Z a; is of x. That it is the same indi- 



X 



vidual function (so far as such a phrase can be considered ap- 

 plicable), or in other words, that it has a better right than any 



other logarithm of ■ — to be considered the logarithm of — 



^ X ^ X 



corresponding to I x, will appear on substituting in I x the re- 

 spective constituents of — for those of .r, that is to say -^^ — ^ 



for j/ and -^ ^ ^^^ ^» ^°^ ^y ^"^^ substitution the expres- 

 sion which I denote by / — will be obtained. 

 Observing that in / — the constituent (see (16.)) 

 I T I , ^ = -^ v//T^, we find 



^ S _1 7/ 



Let I V i/ + z^ = y and ~7=| cos^ , ■ . „ = z", and 



let Z / 9 be denoted by the notation P S, then, by what precedes, 

 we have 



/ (y + v'^ s ) or Px = 1 \/yn^2 



/ 2' -1 y (18.) 



we have also I ( — y — v/ — l -') or Z^ — 



but it is easy to see that 



'"V -T^TT^ = - - ^««o~+ vy?rP ^'''•^ 



Hence P x = l^ ^ + ^/^y -^ -n (21.) 



-i/ 



(19.) 



