178 



off segments from the axes of coordinates, having a certain relation 

 to each other, may be imagined to wrap round or envelope a certain 

 curve, just as we inav see a curve described on paper by the success- 

 ive mtersections of a series of straight lines. Hence there are two 

 distinct modes according to which we may conceive all curves to be 

 generated, namely by the motion of a tracing-point, or the successive 

 intersections of straight lines ; by a pencil or straight edge, as a joiner 

 would say. These conceptions are the logical basis of the methods 

 by which the principles and notation of common algebra are general- 

 ized from the discussion of the properties of abstract number to those 

 of pure space. The former view gave rise to the method of protective 

 coordinates, the latter suggests the method of tangential coordinates, 

 a term which I was the first, I believe, to invent and apply. 



It is sometimes very easy to express both the projection and tan- 

 gential equations of the same curve or curved surface ; it is frequently 

 a matter of extreme difficulty. 



Thus, if the projective equation of an ellipsoid be 



its tangential equation will be 



a, b, c being, as in the preceding equation, the semiaxes. 



Again, if we take the evolute of the ellipse whose equation is 

 =m%, the tangential equation of the same curve is 



I shall not attempt to introduce into this abstract the formulae for 

 the transformation of coordinates, or he several elementary expres- 

 sions which belong to this system, and which must be investigated 

 and known before the method can be used as an instrument of inves- 

 tigation or analogy. My object is rather to give a specimen of the 

 method in the solution of some very difficult problems, and to show 

 how it may be made a powerful instrument of analytical investigation. 



On the Surface of the Centres of Curvature of an Ellipsoid. 

 It is well known to geometers that the lines of greatest and least 

 curvature at any point on the surface of an ellipsoid are at right 

 angles to each other, and that they may be constructed by the inter- 



