82 Mr DAVIES on the Nature of the Hour-Lines 



By proceeding in the same manner, we shall find a repetition of the 

 same results, and in the same order, till another circumference is added to 

 the value of L. Hence the several waves of curve are repeated at regu- 

 lar intervals upon the sphere, in unlimited succession. 



III. 



Since, after 4AD of the abscissa is passed over, the curve returns to a 

 position V, corresponding to that which we have taken as a starting point 

 I, if 4AD be a submultiple of 360, the curve, after one revolution of L, 

 will fall upon that already traced out ; that is, if AD be a submultiple of 



90, otherwise not. If AD be a submultiple of 2, 3, 4, quadrants, 



then, after so many revolutions of the value of L, the curves will return to 

 their original positions, and retrace the same series of waves. 



4 x 90 



If 4 AD, that is be incommensurable with 360, and all its mul- 

 tiples ; or, in other words, if n be irrational, then no such reduplication of 

 the curves can ever take place. In the hectemoria, properly so called, the 

 value of n is, however, always rational. 



It is plain, moreover, that all these branches are equal to one another, 

 being similarly derived from trigonometrical functions, whose successive va- 

 lues at corresponding points are equal through all the mutations of sign and 

 magnitude. If, however, a more detailed proof be required, it is easily ef- 

 fected as follows : 



Let DW be taken (Fig. 1.) equal to DZ, but on the opposite side of 

 D, 



