84 Mr DAVIES on the Nature of the Hour-Lines 



To obtain, then, the ultimate value of cot WDY, take the differential 

 co-efficients of the factors on the right hand side, instead of the factors 

 themselves ; we thus get 



/90 wL 



cos 



cot WDY = cot I.- _ n& ^ LdL i 



dL 

 or tan WDY = n tan I. 



Hence, also, the tangent of the angle which the curve makes with the 

 axis at the point of intersection, is always n times that of the inclination 

 of the equator to the horizon. 



V. 



We shall now discuss the equation ( A D L ) under another aspect, by 



taking the meridian P'AP as the axis, A being the origin as before. This 

 investigation will be more conveniently conducted, by writing the equation 

 in this form 



cos nL = tan D cot I .................. (B_ T ). 



' D,.L' 



We must now recollect that the values of D are not intercepted upon 

 great circles through E and Q, but upon less circles parallel to EQ. Our or- 

 dinates L, therefore, are to be estimated upon these parallels. It will readily 

 appear, that this is but a polar equation of the curve. 



