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. 



Rev. 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, /{x). The first 

 theorem, which is free from signs of integration, implies that 

 between the limits x = a, and x=. a + Aa, 



fi^) = -[ tan ^ tan -^)A=^\ ( 1 ) 



provided that we suppose k a positive quantity, and take the 

 limit to which the second member approaches, as k approxi- 

 mates to 0. When xz=:a, or « + A<^3 the first member of the 

 above equation must be divided by 2 ; and when x 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. : 



"» kdaf{a) 



1 C^ 



in the second member of which the limits -oo and oo may be 

 replaced by any other real limits,/? and q, when all the values 

 of a?, for which y(a;) does not vanish, lie between the limits 

 p and q. This theorem is subject to the same conditions, 



