Right Rev. Charles Graves. 91 



viz., if two spherical conies have the same cyclic arcs, then any arc 

 touching the inner curve will cut off from the outer a segment of 

 constant area. It may be here observed that Bertrand, in his great 

 treatise on the Integral Calculus, attributed the foregoing theorem of 

 Graves to Chasles, who had subsequently arrived at it by an inde- 

 pendent investigation. In a long appendix to the volume Graves gave 

 a method of treating curves on a sphere corresponding to the Cartesian 

 method on the plane, arcs of great circles taking the place of right lines. 

 This theory he worked out in detail, supplying expressions analogous 

 to the fundamental formulae of plane analytic geometry, such as those 

 for tangents, normals, osculating circles, evolutes, &c., and for the 

 transformation of spherical co-ordinates. The whole was the fruit of 

 Graves's independent research, though after the preparation of the 

 Appendix he discovered that Professor Gudermann had partly antici- 

 pated his method, and that the properties of spherical curves had been 

 previously studied by Mr. Davies, who, however, used only polar 

 co-ordinates, whilst those principally employed by Graves were rect- 

 angular. This memoir was greatly admired by Sylvester and other 

 mathematicians, but their high expectations of its fertility have not 

 been fulfilled. 



This was the only mathematical book which Graves published. His 

 other investigations were either embodied in the lectures which he 

 delivered as Professor of Mathematics in the University, or in papers 

 read before the Royal Irish Academy. Having been elected a member 

 of that body in 1837, he filled successively the offices of Secretary of 

 the Council and Secretary of the Academy, and was elected its Presi- 

 dent in 1861. About the same period, Sir William R. Hamilton, 

 McCullagh, and Humphry Lloyd were also members, and the meetings 

 were often made the occasion of announcing the results of the spirit of 

 scientific investigation which then so remarkably prevailed in the 

 University of Dublin. 



While Hamilton was explaining to the Academy in a series of com- 

 munications his new Calculus of Quaternions, several contemporary 

 mathematicians were led to conceive systems more or less analogous to 

 his, and like it, involving new imaginaries. Graves proposed a 

 system of Algebraic Triplets of this kind. / But it must be said of it, 

 as of the other similar systems, that it could not lay claim to anything 

 like the power and flexibility of the Quaternions, and was not, indeed, 

 so much a working method as an interesting mathematical curiosity. 

 Other papers of his, published by the Academy, related to the theory 

 of differential equations, to the solution of the equation of Laplace's 

 functions, and to curves traced on surfaces, particularly on surfaces of 

 the second degree. He gave a simple geometrical proof, published 

 also in ' Crelle's Journal,' of Joachimsthal's theorem, viz., that at all 

 points of a line of curvature on an ellipsoid, the product PD is constant, 



