upon the Antique Dials. 101 



The first of the values of v in Case II. is that adapted to a hectemorial 

 conical dial, such as the Choragic is stated to be. However, the discussion 

 of the applications of these formulae must be reserved till a future occasion, 

 when we devote our attention particularly to that part of our subject. The 

 values of v f , too, in these several cases are so similarly obtained, that any 

 details will be unnecessary, and we shall merely set them down. 



Case 1. v' = a . cosec X sec X cos p cos v cosec ( + p) cos L. 

 i/ = ia sec* X sec* -* 



p =ta 



2 i =: , 



Case2 -' g> 2 *L 



sec z X cosec 2 -5- 



Case 3. ' = %a . cosec X sec X sec ( + nL) sec ( iL) cos nL. 



XIX. 



It is plain, that if for L we write cosec *. . 0, we shall have the equations 

 of the hectemoria, when developed upon a plane : but it will, upon the 

 whole, be more convenient, for the sake of general investigation, to put 

 n cosec \ =. m, and we get the developed curve in an equation of the same 

 form as that which represents the curve upon the conic surface itself. It is 

 unnecessary to repeat them ; and we proceed to ascertain their forms. 



XX. 



For this purpose we resume the first general form, viz. (C D)t ) found 

 in XIV. 



cosec 2 X cot I 

 ' cot X cot I + cos m L' 



merely putting m for n cosec X, to adapt it to the development aimed at. 



There are several cases, according to the values and signs given to cot *. 

 cot I. 



VOL. XII. PART I. O 



