106 



Mr DAVIES on the Nature of the Hour-Lines 



values of the nine cosines above mentioned may then be formed into a 

 tablet, as below. 



The several values of x, y, z, then, whilst the origin remains unaltered. 

 will be 



x of cos I sin X + y sin I -f z' cos I cos X. 

 y = a/ sin Z sin X y cos Z -f- z 1 sin Z cos X. 

 2 = a/ cos X + ^ sin X. 



If we substitute these values of x, y,z in the equation (A.ryx) we shall 

 obtain the sheet of the hectemorial cone referred to the same origin, but 

 new axes, a/,y',z f ; but we may refer them to any other origin by simply 

 adding to the values of x, y> z the co-ordinates of that point. The object 

 we have in view, taken in connexion with the form of the equation (A xy z) 

 requires that the addends to x and y should be both symmetrical and simple. 

 There are four points which, under this aspect, offer themselves to our con- 

 sideration, viz. the point of contact of the plane and sphere, and the three 

 intersections of that plane with the axes of x, y, z. 



The point of contact offers symmetrical results, its addends being 



/" = a cos x sin ?, 

 a." = a cos A cos I ; 



