X. On Continuity. 



By Professor Horace Lamb, M.A., F.R.S. 



Received and read March 9th, 1897. 



A word or two of apology may possibly be needed 

 for bringing before the Society a paper whose most 

 definite results consist merely in new, or rather modified, 

 proofs of one or two fundamental theorems in the Infini- 

 tesimal Calculus. It may, perhaps, be allowed to urge 

 in excuse that although in the past English mathemati- 

 cians have taken their share in discussions concerning 

 the logic of the mathematical sciences, yet the particular 

 subject at present in view, the Theory of Functions of a 

 Real Variable, has as yet attracted very little notice in 

 this country. It may also be said that the current proofs 

 of the theorems referred to present little variety, and that 

 the five or six treatises in which they are to be found 

 reproduce the same steps in almost exactly the same 

 order. There ma}-, therefore, be some justification for 

 an attempt to bring these theorems more immediately 

 into connection with first principles. Of any fundamental 

 novelty in the treatment of the matters involved there 

 can, from the nature of the case, be no question ; but 

 opportunity is taken of insisting on a point which, simple 

 as it is, is in some danger of being overlooked in the 

 modern endeavours to establish the Calculus on a purely 

 numerical basis. 



1. It is necessary to explain, in the first place, the 

 point of view from which the theorems referred to are 

 developed. For reasons to be given further on, the 



May 20th, 1897. 



