upon the Antique Dials. 109 



XXIV. 



By giving to the quantities A, /, n, various values, to adapt them to dif- 

 ferent dials, and to the several hour-lines upon each, we shall be able to de- 

 termine for t a corresponding value of r, and thus assign the assemblage of 

 points which constitute the several hectemoria. Though it is not my inten- 

 tion in the present paper to enter upon the discussion of particular dials, 

 yet it seems necessary to enter a little into discussion of the forms assumed 

 by our equations when particular values are given to A and /. It is neces- 

 sary, both in illustration of the formula, and in order to divide the whole in- 

 vestigation into parts as nearly equal as is practicable. 



1. Let x = ; then for all values of I we shall have sec x = 1 ; cosec 

 x = infinity ; .and 



_ a sec. 6 sec cosec . . 



~~ tan ( tan I sec cos n cos" 1 sin I + 1 



This shews that the origin is infinitely remote, as from other simple 

 considerations we know that it should be. The method to be employed in 

 this case will be presently explained. 



2. Let I = 0, and 1= 180, whilst x is arbitrary. Then sin I = ; cos 



/=1; and 



a sec A cosec x sec t 



r = 



it sin A 

 { sec A tan I cos n cos 1 1- 1 



e 



This is the equation of all north and south dials, whatever be their in- 

 clination to the horizon. 



3. If in (2) we put x = I, we shall obtain the equation of the vertical 

 south dial, such as the NOTOS* of ANDRONICUS CYRIIHESTES, figured in 

 plate x. vol. i. of STUART and REVETT'S Antiquities of Athens. 



* It would have been more conformable to the common usage to have designated 

 these dials by names diametrically opposite upon the compass ; but if, in all cases, we 



VOL. XII. PART I. P 



