114 Mr DAVIES on the Nature of the Hour-Lines 



Let them be referred to the point of contact as origin, that is, the east- 

 ern dial to the east point of the horizon, and the western to the west. 

 Then x = 0, / = r90"; the addend to y is a, and x and z, each 0. 

 Hence the values of x, y, z, in (XXII.) become, 



a 



and the expression is reduced to 



/ - ~~ l 



af=i v y 2 + a 2 cos cos y 



v y + 



which is the equation of the hectemorial curves upon the dials touching the 

 sphere at the east and west points of the horizon, and referred to the rectan- 

 gular axes of, y'l 



But since, from the method of transformation that we have employed, 

 the axis of y' is the intersection of the dial with the equatorial plane, it 

 readily occurs to us that the better way will be to interchange the #' and y' 

 at the same time that we cancel the accents. Hence 



_ i -{- x 



y =r s/ x' 2 + a 2 tan I cos n cos / .... I A. \ 



Jx*+a* \ e , ) 



A further transformation will greatly facilitate the practical calculation of 

 these dials. Let <? = tan" 1 x, (rad. sphere = a) ; then 



V X s + a 2 = sec <p 

 y a tan I sec <f> cos n cos" 1 sin p 

 = a tan I sec f> cos n (90 p), or, restoring tan'ir, 

 = a tan I sec tan ~ l x cos n (90 tan -'a?) ..... / A , ,\ 



Using HUTTON'S tables, where the natural and logarithmic functions of 

 the angle <p stand at the same opening, we find the value of y with a single 

 opening of the book, so long as we suppose 9 to vary from step to step by 

 some exact number of minutes a degree of precision which may be deemed 

 greatly more than sufficient. 



