184 
From Fourier’s theorem the author finally deduces a 
Ae) 
Zt 
—, ViZ.: 
theorem for the discontinuous expression 0 
= some OS is A dadv dw cos (@ —x)vo—to+ =e" fe), (4) 
@ 
in which the symbols | are both used in a limiting sense. 
0 
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 formule 
(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 « 
enters into the second member exponentially, which affords 
facilities for both the direct and the inverse processes of the 
differential calculus. 
In the fourth theorem the author remarks that both # 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 haying 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 Reale Academia della Scienza di To- 
renzo. Serie Secunda. Tomie. VI. Presented by the Academy. 
