249 



ciation to the above theorem. For let s denote the arc of the 

 hyperbola, 



measured from the vertex to the point of which the ordinate 

 (y) is \/{a^ - b^) ; and let s be the arc of the ellipse, 





a~ + b^ b^ 



counted from the extremity of the lesser axis to the point of 



b^ 

 which the ordinate is — r ; and we will have 

 a' 



Sii S^i+i + S2i Sgr+i = Tr{a + (is + ys'} , 



Oj |3> y, being, as before, algebraic functions of a and b. 



" By making i = i' = 0, we light on the theorem which I for- 

 merly communicated to the Academy. In this case y, in the 

 last equation, vanishes. 



*' The above theorem, which I think there would be some 

 difficulty in establishing directly, I have demonstrated very 

 simply, by employing the method of elliptic co-ordinates, with 

 the vast power of which, as an analytical instrument, all 

 readers of the mathematical publications of the present day 

 must be familiar. I subjoin a sketch of my proof, which, I 

 trust, may not be altogether uninteresting. 



** In the first place, I calculate the values of 821, /Sgf+i, Sgi, 

 Sg.Mj by means of a general formula which I obtained some 

 years since, and which has appeared in the Proceedings of 

 the Academy. Then, after a few easy transformations, I 

 am enabled to write the sum of the two products, S-a S-n^i and 

 S-iiSo.M, as a double definite integral, which, although con- 

 sisting of a variety of terms, is found ultimately to depend on 

 two only, which are distinct and independent. Of these, one 

 expresses the superficial area of a certain ellipsoid, and the 

 other, the sum of all the superficial elements of the same 

 ellipsoid, divided by the squares of the areas of the sections of 



