116 Mr DAVIES on the Nature of the Hour-Lines 



expression for the " polar dial *," either from the same special method 

 there employed, or as a case of XXVIII. The latter is the readiest. 



Here, then, I = o, (or /= 180 for the other intersection) ; and therefore 

 ocf sin / = o, a cos I = a ; 



+ a 2 cos n cos 



t 



>J x* + a a 



But if tan ""'a?' = <p, there will result , = cos ? ; and, dropping ac- 



V JC }"# 



cents, 



y = a tan I sec <p cos n cos 1 cos <f 

 = a tan I sec p cos n p 

 = a tan I sec tan ~ ! x cos n tan -1 <r. 



The first branches of these hectemoria are traced by Mr CADELL f, 

 which, though less accurate than his other drawings, are yet sufficient to 

 give a general notion of their character. They seem to he sketched without 

 much instrumental assistance, or reduced by the eye from larger projections. 



XXX. 



It has already been stated, that the hectemoria being traced upon a 

 sphere, and projected upon a cylinder touching it at the equator, is but a 

 modification of the harmonic curve J. The equation of the curve upon the 

 developed cylinder (the axis of x being the unrolled equator, and y the cor- 

 responding value of tan D), is reduced to 



y = i cos nx ; or when referred to radius a, 



/ x\ 



ij =r ia cos n I I 

 * \a/ 



Put = a' or n a' = a ; then y =s n a', i. cos ( ) 

 n \ na / 



^. n a. cos x. 



* That is, a dial which touches the hectemorial sphere at the intersection of the 

 equator and meridian of the place for which it is made. 

 f Plate II. fig. 3. of the Memoir already referred to. 

 Art. XIII. 



