88 Mr DAVIES on the Nature of the Hour-Lines 



VIII. 



The origin may be transposed to any point on the surface of the sphere, 

 but the formulae to which such a transfer gives rise, are generally too com- 

 plicated to be of much use. Fortunately, however, the transformations of 

 this kind, which our present object requires, are extremely simple, being only 

 to points in the equator, and to the equatorial poles. 



The origin is transposed to any point in the equator whose longitude is 

 given (ipL/), just in the same manner as in rectangular co ordinates, by 

 simply annexing this quantity to L in the general equation (A ). We 



thus obtain 



tan D = tan I cos n(L I/) ......... (^ 



Again, it may be transposed to the equatorial poles in the following 

 manner. Let 



D' = 90 D; then 

 tan D = cot D'; 



or, taking the reciprocals of these, we shall reduce to 



1 1 I 



tan D ~~ tan I ' cos nL' 



tan IX = cot I sec nL ............ (-^TYI ) 



The expression (A D , jj just obtained is in fact only a polar equation of 



the curve ; and it is easy to see how it might have been primarily derived 

 from first principles, and the other equations from it, in an order exactly the 

 reverse of that which we have followed. The equation is, moreover, one of 

 the most important of all in our inquiry. 



This equation reckons its angles L from the meridian of the place, 

 PAP' ; but it might be referred to any other meridian whose longitude is 

 I/ as before. Then, 



tan !> = cot I sec n(LL') ......... ( A rj'L:L,)- 



