12 REPORT 1871. 



by takino- _ =1— i (in which 2«> 1, so that m may have any value except such 



as lie between 1 and -1), and using the relation r(«)r(l-a) = 7rcosec«n-, we 

 obtain 



j 8ina?'"«fo = r|l+ - jsin^. 



Similarly, by integrating J^ j"^ e-^"y coai/d.v dy, we find 



Jo 



■•00 



TT 



cosx'^dx=r( l-\ Icos, 



^ ' mJ 2»i 

 (tn between 1 and oo). 

 The author had calculated a Table of the values of I smx"dx, \ cosx^dx for 



different values of x ; and the cui'ves y = J sin a^da, y =J cos a^da, as obtained from 



them, were drawn and exhibited to the Section, the discontinuities in each being 

 remarked. [The Tables and curves will be found in the ' Messenger of Mathema- 

 tics,' 1871.] 



On Lamberfs Proof of the Irrationality of tt, and on the Irrationality/ of cer- 

 tain other Quantities. By J. W. L. Glaisher, B.A., F.R.A.S. 



The arithmetical quadrature of the circle, that is to say, the expression of the 

 ratio of the circumference to the diameter in the foi-m of a vulgar fraction with 

 both numerator and denominator finite quantities, was shown to be impossible by 

 Lambert in the ' Berlin Memoirs ' for 1761 ; and the proof has since been given in 

 an abrido'ed and modified form by Legendre in the Notes to his ' Elements de Geo- 

 metric.' Although Legendre's method is quite as rigorous as that on which it is 

 founded, still, on the whole, the demonstration of Lambert seems to afford a more 

 striking and convincing proof of the truth of the proposition ; his investigation, 

 however, is given in such detail, and so many properties of continued fractions, now 

 well known, are proved, that it is not very easy to follow his reasoning, which ex- 

 tends over more than thirty pages. The object of the present paper is to exhibit 

 Lambert's demonstration of this important theorem concisely, and in a form free 

 from unnecessary details, and to apply his method to deduce some results with 

 regard to the irrationality of certain circular and other functions. 



The theorem which Lambert proves, and from which he deduces the irrationality 

 of TT, is that the tangent of a rational arc (i. e. an arc commetisurahle with the raditts) 

 must he irrational; and this he demonstrates by means of an expression for the 

 tangent as a continued fraction, viz. 



I X X Ji X" X" /*•>. 



tan - = — -_, (i) 



adopting an established notation for continued fractions in which that which fol- 

 lows each minus-sign is written as a factor, to save room. 

 Consider a continued fraction 



^. /3, ^3 . . . (ii) 



«,+ «,-t- <«3 + &C.' ^ ' 



and let — be the wth convergent to it ; then we know that 

 ^^ 



and 



PnJ\-X_. _y_, ^Sl...^n , (i;;^ 



* These results can easi]^ be proved by induction. 



