350 Mr. J. W. L. Glaislier on [Nov. 20, 



Let e~ a * = q n , 



71 being a positive integer, whence 



and 



j°V* 2 e | 2aj^— —J j ^=i7ri(l-2^ +1 -22 4( " +1) + 2 2 9(H+1) - &c.) 



) 



_^ / „ + iK w+1 J (74) ? 



whe^e K w+1 are what & and K become when q is replaced by g'" 4 " 1 , 

 so that k n+l is the modulus obtained by the first real transformation of 

 the (n + l) th order. 



As a particular case put n=l, and the formula (74) gives 



^ e~**e 1 2a^— J j <fe= (JEtf*)* .... (75). 

 To evaluate the other integral, multiply the equation 



•g-^2Ka^_2^ g - n — sin 3aaj+2g ,a * sin 5er#— &c. 



by e~ x<2 and integrate, replacing the integrals on the right hand side 

 by their values from the formula 



xe x 



Jo 



we thus find 



J *i*r**B. (^^dx=^a((fe^* - Sqh-^ 2 + &q?eT?« - &c.) 



Put 5 as before, e~ a2 =q n , 

 n being a positive integer, then 



--m 



and the equation becomes 



But 2tf-6tf+10q?- &c.= ( / ^ / (^) 3 } 



