29 -t lioyal Irish Academy. 



to knowledge which you have brought back. Any of us who contem- 

 plates the circle at the base of the cone passing successively into the 

 ellipse, the parabola, the hyperbola, as the sectional plane shifts 

 through its varying predicaments, can well conceive how the Sine, 

 Cosine, and other Cii'cular Functions may have their modified counter- 

 parts in expressions applicable to these and similar transformations ; 

 and can, in some measure, appreciate the great achievement of Le- 

 gendre in discovering and generalizing the correlative group of the 

 Elliptic Functions. It is not yet sixty years since mankind profited 

 by this most valuable accession to the instruments of analysis. Abel, 

 carrying Legendre's investigations forward into a region of expressions 

 still more complex, soon after gave to view the vast outlying 

 province of the Abelian or Hyper-Elliptic Functions. These In- 

 tegrals, amongst the highest conceptions yet grasped by human 

 intelligence, had a necessary affinity to the discoveries of Legen- 

 dre ; but the links of their relationship were wanting, and the unity 

 which is the soul of the law remained undemonstrated. It was 

 then that Jacobi, taking up the torch which had fallen from the 

 hand of the youthful Abel, and exploring the intermediate abyss 

 with a still keener scrutiny, had the happiness to discover and demon- 

 strate one of the lines of affinity connecting those transcendental 

 properties of space and figure with the Elliptic Functions of Legendre. 

 Tou, Professor Malet, have not been less happy in demonstrating — as 

 the presence of the mathematicians around you attests — with equal 

 certainty and with a farther reach of vision, that not one only, but all 

 these roots of the Hyper-Elliptic system, connect themselves, through 

 retroverse transformations, which you have shown need not exceed 

 four in number, to the precedent theory of Legendre, as that to the 

 primary conceptions of Euclid and Plato. I may compare you two to 

 the listeners to the inland murmur of the sea-shell : Jacobi to Landor's. 

 to whom — 



" It remembers its august abodes, 

 And murmurs as tbe ocean murmiirs tbere." 



You, to Wordsworth's, to whose ear it has spoken more articulately — 

 if I may, without profanation, so far paraphrase his words — 



' ' Of central la-w subsisting at the heart 

 Of endless transformation." 



