upon the Antique Dials. 115 



XXVIII. 



In the same manner, but with a rather less simple result, we might ob- 

 tain the equation of the hectemoria upon any plane parallel to the earth's 

 axis. We might also, by the formula of EULER, attain the same object ; 

 and it will be found upon trial to be in exactly the same form. 



For, in this case, our x and y coincide with his y and x respectively. 

 Denoting, then, by t the trace of the equatorial upon the dial plane, and 

 , taking the other quantities in the usual manner, we obtain by EULER, 



x, of 90 I, and x y, of y = 90 

 hence x = of sin I + a cos I 

 y = x' cos I + a sin I 

 z=yf 

 .: x* + y* = (a + tf') 2 and the hectemorial equation becomes, 



, / T- - 1 of sin I + a cos I 



y = t v or + a* cos n cos , . 



J x 3 + a 3 



Or, if we put a cot / = b, it is 



,5 5- ~ l (x+b) sin I 



y = ^ *J a* + a 8 cos n cos . . 



v x + a 



= a tan I sec tan ~ l x cos n cos- 1 ! 2-i sin I cos tan ~ l x t A \ 



Making the interchange already noticed, and dropping the accents from 

 a/, yf, we shall obtain the result from the method of M. FRAN AIS. It 

 must be recollected that / is the longitude of the point of contact of dial with 

 the hectemorial equator. 



XXIX. 



We have considered the east and west dials, and obtained for them a 

 simple expression in XXVII., and we can in the same manner obtain an 



