184 

 From Fourier's theorem the author finally deduces a 

 theorem for the discontinuous expression of -— -, viz. : 



•i^^=: — -^V V \ dadvdwcos{(a—x)v—tw + --]w /(a), (4) 

 r 'rr(»Oj-QoJo Jo \ ^J 



r** ... 



in which the symbols \ are both used in a limiting sense. 



Jo 

 This theorem, it is observed, is susceptible of important appli- 

 cations, in the theory of definite multiple integrals. 



As respects the expression of discontinuity, the formulae 

 (1), (2), (3), are shewn to be equivalent. The advantage 

 which the second possesses over the first is, that x enters into 

 the second member in a rational form ; and the advantage of 

 the third, or Fourier's theorem, over them both, is, that x 

 enters into the second member exponentially, which affords 

 facilities for both the direct and the inverse processes of the 

 difi'erential calculus. 



In the fourth theorem the author remarks that both x and 

 t enter exponentially. 



Sir William R. Hamilton made some observations on Mr. 

 Boole's communication. 



Professor Harrison made some remarks on the peculiarities 

 of the anatomy of the Emu. [This paper, not having been 

 received in proper time for insertion here, will be printed in 

 the Appendix, No. VII.] 



Professor Allman announced the addition of Deotis Mari- 

 tima to the Irish Fauna. This plant he had observed during 

 the latter part of the summer of 1845, in the sand-hills in the 

 neighbourhood of Dungarvan, county of Wexford. 



DONATIONS. 



Memorie della Beale Academia della Scienza di To- 

 renzo. Serie Secunda. Tome. VI. Presented by the Academy. 



