V1—On the Analysis of Discontinuous Functions. By GrorcE Boor, Esq, 
Read 20th July, 1846. 
THE design of this paper is to illustrate the mathematical doctrine of discon- 
tinuity, and to present a few remarks on some connected subjects. We shall 
first deduce in succession three theorems, by which the function /(#) may be 
expressed in subjection to any proposed form of discontinuity. The last of these 
theorems will be identical with Fourier’s, and the second, and probably the first, 
are known; but the connexion in which they will be presented is, perhaps, new, 
and may not be devoid of interest. Subsequently, a theorem will be deduced for 
the discontinuous expression of = which admits of very interesting applica- 
tions to the theory of definite multiple integrals. One such application will form 
the subject of another paper. As respects the theorem of Fourier, usually ex- 
pressed in the form 
(a) — = {) dadv cos(av—av) f(a), (1) 
we shall endeavour to show that the right hand member is to be considered as 
nothing else than an abbreviation, sanctioned by custom, of the expression 
limit of y ‘ {dade «cos (av—wzv) f(a), (2) 
TT J—xwJ0 
k approaching through positive values to 0; and that if this meaning is ne- 
glected,—if the sign of integration relative to v is taken in its ordinary import, 
we have no assurance of the truth of the theorem. It might appear to be super- 
fluous to insist on a position like this, on which the testimony of some writers is 
