upon the Antique Dials. . 105 



means of the common equations for the interchange of rectangular and po- 

 lar co-ordinates. Thus, a being the radius of the hectemorial sphere, and 



putting tan I = i, 



z = a sin D, 



y = a cos D sin L, 



x = a cos D cos L, 



from which tan D = _. , and cos L = 



Jx 2 + y z v x + y 



Then substituting these values in (A D)L ), we obtain 



= icOSW COS" 1 , (Ajcyz) 



which is the equation of the hectemorial cone. 



XXII. 



Our object being to find the intersection of this cone with a given arbi- 

 trary plane, the most obvious method of proceeding is to so transpose the 

 origin and direction of the co-ordinate axes, that two of them shall lie in 

 that plane, and the third be at right angles to it. The method of EULER, 

 which is commonly employed for this purpose, and which is often found to 

 be the most simple, both in its application and results, is, in the present 

 instance, less convenient than the symmetrical formulae of M. FiJANqAis. 

 I shall accordingly employ the latter method in the general transformation. 



Let a, a', a" be the cosines of the angles made by ocf ^ . i 



/ witn x f y, *., 



- b > b '> b " ^ i spectively. 



. . c, d, c" Vv -,: . . . i.sfj 



Then, draw OP perpendicular to the plane of 

 projection, meeting the sphere in P, a point 

 whose latitude is A, and longitude /; and let this 

 be taken as the axis of z'. Draw OP' in the 

 plane of the meridian zPz' perpendicular to OP, 

 which take as the axis of x'. Lastly, draw OK 

 perpendicular to POP', for the axis of y'. The 



* Journal de 1'Ecole Polytechnique, cab. xiv. pp. 182 190. 



re- 



