4 SIXTH REPORT. — 1836. 



In prosecuting further inquiries it will be desirable to considef 

 which are the forms of transcendents which ought to be reckoned as 

 next in order to those whose properties have been hitherto most in- 

 vestigated, viz., to the Circular Logarithmic and Elliptic functions. 



It appears to me that the transcendents might be divided and 

 classed according to the number of them which it is requisite to com- 

 bine in order to obtain an algebraic sum. Thus the transcend- 

 ent /— r is of a more complicated nature than / 



J Vl_+A'' J Vn-^* 



because it is requisite to unite three terms of the former to obtain an 

 algebraic sum, while it suffices to add two terms of the latter one. 



According to this view the transcendent / — ^ will be the 



representative of a class whose properties are to be examined by 

 themselves, and which are probably irreductible to transcendents of 

 a lower class. Before, however, occupying ourselves with these, it 

 is well to inquire what results these new methods give when applied 

 to the arcs of the Conic Sections, a subject which was supposed to 

 have been almost exhausted by the labours of Legendre, but which 

 the researches of Jacobi, Abel, and others have shown to be far from 

 being so. 



I have found, with respect to my own method, that besides the 

 theorem which I originally met with concerning the sum of three 

 arcs of the Equilateral Hyperbola, it likewise gives a number of other 

 theorems respecting the sums of the arcs of the Conic Sections. 



But which of all these theorems are essentially different from each 

 other it requires much time to thoroughly examine. And since it is 

 desirable for the sake of verification, and to avoid falling into errors, 

 to accompany the processes of analysis with numerical examples, 

 these examples, if calculated to six or seven places of decimals, often 

 run into extreme prolixity, and would be best accomplished by the 

 assistance of several independent calculators. 



On the Calculus of Principal Relations. By Professor 

 Sir W. R. Hamilton. 



The method of principal relations, of which Sir W. Hamilton 

 gave a short explanation, is still more general than the analogous 

 researches in optics and dynamics presented to former meetings of 

 the Association. By it the author proposed to reduce all questions 

 in analysis to one fundamental equation or formula, no matter how nu- 

 merous the conditions, or the independent variables might be. He 

 has found the following relation, which he has termed principal, to 

 subsist between all differential functions, no matter how numerous, or 



independent the variables, viz. : — — - =-r — 



S d X a X 



