49 



series expressed by it, commences, in all cases, with a finite 

 quantity, such that the terms of the series, by continual dimi- 

 nution, tend to zei'o. 



The circumstances are very different with respect to 

 p _ 2'^g + 3V— 4V'+ &c. As observed in the foot-note at 

 p. 44, the commencement of convergency, in the limiting case, 

 is at a term infinitely distant from the origin of the proposed 

 series, and infinitely great. What analogy can there be be- 

 tween the general converging series — if it may be so called — 

 of which this is a limiting case, and ordinary convergent se- 

 ries? And can it be affirmed, of any one of its cases, that the 

 terms necessarily tend to zero ? The answers to these ques- 

 tions will, I think, destroy all idea of analogy in such examples 

 as those adduced above. 



1 have been compelled, in several parts of the present 

 Paper, to dissent from certain doctrines and opinions promul- 

 gated by some very distinguished writers on analysis. In de- 

 veloping the principles and views here submitted to the Royal 

 Irish Academy, I could not easily avoid a reference to these. 

 I trust, however, that I have done so in no captious or un- 

 candid spirit : I have only been anxious to arrive at truth in 

 an inquiry of acknowledged perplexity, and of interest, perhaps, 

 in the estimation of some, sufficient to justify the attempt. 

 There are one or two points of analytical delicacy involved in 

 this inquiry, which may perhaps be open to further discussion : 

 if I have myself fallen into error in my treatment of these, I 

 hope 1 shall be indulged with the same candour and conside- 

 ration which I have endeavoured to exercise towards others. 



Professor Mac Cullagh made a communication on the 

 subject of Total Reflexion. 



In the case of total reflexion the vibrations which take place 

 in the rarer medium are in general elliptical, and when this 

 medium is a crystal, the equations by which the ellipse of vi- 

 bration is determined are very complicated. The projection 



