Pdx 

 Prof. J. R. Young on the Integral I — . 11 



FGH ; but as the length of these filaments was then consider- 

 ably reduced, their tendency to twine themselves round A, from 

 any sudden motion of the eyeball, was diminished, and there 

 has been no enlargement of the black spot or Musca during 

 the years 1846 and 1847. 

 St. Leonard's College, 

 December 11, 1847. 



/'dx 

 1 1. 0)1 the Integral I - — , and o?i some consequences that have 



been deduced from it. By J. R. Young, Professor of Ma- 

 thematics in Belfast College"^. 



IT is the object of the present short paper to remove some 

 obscurities connected with the ordinary treatment of the 



/ dx 

 simple integral / — . It will be anticipated therefore that the 



remarks which I have to offer are of a very elementary cha- 

 racter: too much so indeed to entitle them to a place in a 

 Journal of this kind, were it not that such obscurities, in the 

 first elements of science, as experience has often shown, are 

 frequently the source of important errors in its more recondite 

 applications. 



The integral just adverted to is a particular case of the more 



general form I x^dx, of which the value, disregarding cor- 



.r" "*" ' 



rection, is known to be . Jn the particular case noticed, 



n ->rl 



that namely in which «= — 1, this expression for the value is 



said iofail; though it is admitted to be valid in every other 



case, whether n be whole or fractional, positive or negative. 



It may possibly be remembered by some of ihe readers of 

 this Journal, that such an isolated lailing case, in a general 

 algebraic formula, is an occurrence that I have endeavoured 

 to show can never happen ; and that if any such formula hold 

 for all values between a and b, it must e(]ually hold for the 

 extieme limits a and b themselves. 



Ill the instance before us, the particular information, which 

 the general form is supposed to fail in supplying, is obtained 

 from other considerations ; and the value of the wanting inte- 

 gral afKrmed to be log x. And the reason sometimes assigned 

 for the inefficiency ol the general form, in the particular case, 



/dx 

 — = log Xi supposes the function of 



* Communicated by the Author. 



