302 



each summation being performed with respect to an auxiliary integer m, 

 from m = 0 to m = 1 . 



6. Accordingly, without using imaginaries, it is easy to prove that 

 this expression (5) satisfies all the recent conditions (3), and is therefore 

 a correct expression for the partial sum 



S n,rV_ 



while a similar proof of the recent equation 0 = &c. 



7. But to form practically, with the easiest possible arithmetic, a 

 Table of Values of s, for any given period, p, we are led by No. 3 to 

 construct a Scheme, such as the following 



Table op Values oe s 



n, 





r — 5 



4 



3 



2 



1 



0 



Verification. 





5 = 1 



0 



0 



0 



0 



1 



2s = l 



1 





0 



0 



0 



1 



1 



2 



2 





0 



0 



1 



2 



1 



4 



3 





0 



1 



3 



3 



1 



8 



4 





1 



4 



6 



4 



1 



16 



5 



2 



5 



10 



10 



5 



2 



32 



6 



7 



15 



20 



15 



7 



7 



64 



The President read the following paper by the late Sir "William R. 

 Hamilton : — 



On a JSTew System oe Two Geneeal Equations oe Curvature, 



Including as easy consequences a new form of the Joint Differential 

 Equation of the Two Lines of Curvature, with a new Proof of their 

 General Pectangularity ; and also a new Quadratic for the Joint 

 Determination of the Two Radii of Curvature : all deduced by Gauss's 

 Second Method, for discussing generally the Properties of a Surface ; 

 and the latter being verified by a Comparison of Expressions, for 

 what is called by him the Measure of Curvature. 



1. Notwithstanding the great beauty and importance of the investiga- 

 tions of the illustrious Gauss, contained in his Disquisitiones Generates 

 circa Superficies Curvas, a Memoir which was communicated to the Royal 

 Society of Gottingen in October, 1827, and was printed in Tom. vi. of 



