182 
In conclusion, Mr. Smith submitted that it would be 
desirable to have careful drawings, and, where practicable, 
rubbings also, made of all such existing monuments, in order 
that these most interesting memorials, which contain valuable 
confirmations of written documents, as well as curious illus- 
trations of the manners and customs of bygone times, may be 
preserved from oblivion; and stated that he would be much 
gratified by receiving any communications on the subject, 
though they went no further than to state the existence of 
such crosses, in order to complete the materials for a general 
history of these Christian memorials, so deeply interesting, 
even in an historical point of view alone. . 
Rey. Charles Graves, F.T.C. D., read a Memoir, by Mr. 
George Boole, of Lincoln, on Discontinuous Functions. 
The author deduces in succession three theorems for the 
expression of the discontinuous function, f(z). ‘The first 
theorem, which is free from signs of integration, implies that 
between the limits «=a, and z=a+ Aa, 
fa) = 7 (tan EAE —tan 7) fa), (1) 
provided that we suppose & a positive quantity, and take the 
limit to which the second member approaches, as k approxi- 
mates to 0. When «=a, or a+ Aa, the first member of the 
above equation must be divided by 2; and when @ transcends 
those limits, the first member is to be replaced by 0. From 
this formula, the author deduces his second theorem, involving 
one sign of integration, viz. : 
1C? | kda fla) 
sa ta he + (a—a)” @) 
in the second member of which the limits -« and may be 
replaced by any other real limits, p and g, when all the values 
of x, for which f(x) does not vanish, lie between the limits 
pandg. This theorem is subject to the same conditions, 
