188 



. a 



. . . (17) 



r 



The foregoing propositions will give some idea of the fertility of 

 this method, and of the ease and simplicity of its application. I 

 propose to develope it systematically in the memoir, of which this is 

 merely a specimen. Many other systems of coordinates may be 

 imagined, such as the parallel system of Ohasles, or the curvilinear 

 ordinates of Lame ; but it may be questioned whether there is any 

 system so directly reciprocal to the Cartesian method as this of Tan- 

 gential Coordinates. 



Note. Since the above abstract was written, my attention has been 

 drawn to the results of an elaborate investigation of the protective 

 equation of the surface of the centres of curvature, by the Rev. G. 

 Salmon, Fellow of Trinity College, Dublin, and published in the 

 Quarterly Journal of Pure and Applied Mathematics of Feb. 1858. 



Although this surface has been familiarly known to the continental 

 mathematicians since the time of Monge, none of them have ventured 

 to grapple with the enormous difficulties which stand in the way of 

 exhibiting its protective equation, or its equation in xyz. These 

 difficulties have been surmounted by Mr. Salmon ; and the resulting 

 equation, which is of the twelfth degree, contains no fewer than 

 eighty-three terms. 



