344 



Prof. Cayley on the Theory of the Evolute. 



the following 



pure sulphur was removed from this solution, and then ammonia 

 added to precipitate the alumina and sesquioxide of iron. These 

 were not separated, but after washing were ignited, and afforded 

 a weight of 042 gr., equivalent to 2*94 per cent. ; and sub- 

 tracting from this the 1*75 per cent, alumina previously deter- 

 mined, we have 1*19 per cent, as the amount of sesquioxide of 

 iron present in the mineral. 



The results above, when tabulated, will give 

 percentage composition : — 



Fluoride of calcium . . 8*01 

 Chloride of calcium . . 0*16 

 Lime ....... 41'03 



Magnesia ...... 0*12 



Alumina 1*75 



Sesquioxide of iron . . . 1*19 

 Phosphoric acid .... 44*12 

 Sulphuric acid .... trace 

 Carbonic acid .... 0*40 

 Insoluble matter . . . 1'41 

 Water 1*44 



99-63 



XLIX. On the Theory of the Evolute. 

 By A. Cayley, F.R.S* 



ACCORDING to the generalized notion of geometrical mag- 

 nitude, two lines are said to be at right angles to each 

 other when they are harmonics in regard to a certain conic called 

 the Absolute ; this being so, the normal at any point of a curve 

 is the line at right angles to the tangent, and the Evolute is the 

 envelope of the normals. 



Let the equation of the absolute be 



6 = («, b, c,f, g, KJw, y, zf=0, 

 and suppose, as usual, that the inverse coefficients are (A, B, C, 



F, G, H). Consider a given curve U = (*X^ V> z ) m== ®> an( ^ 

 suppose, for shortness, that the first differential coefficients of U 

 are denoted by L, M, N. Then we have to find the equation 

 of the normal at the point (%, y, z) of the curve U=0. 



The condition that any two lines are harmonics in regard to 

 the absolute, is equivalent to this, viz. each line passes through 

 the pole of the other line in regard to the absolute. Hence the 

 normal at the point [x, y, z) is the line joining this point with 

 the pole of the tangent. Now, taking (X, Y, Z) as current co- 

 * Communicated by the Author. 



