100 Mr DAVIES on the Nature of the Hour-Lines 



too, it is well understood by all who have attended at all to the practice of 

 such operations, that, generally, those are the most easily constructed in 

 which the fewest products or quotients appear. Yet as the method is easily 

 derived ; and, moreover, as in one particular case (and that the most import- 

 ant, the Choragic Dial at Athens being an alleged specimen) the formula 

 possesses considerable simplicity, it is worth while to make the trans- 

 formation. 



Either cot x cot I is numerically greater than 1, or it is not. 



CASE I. Cot A. cot I ^ 1. Then put * = tan" 1 cosnL, and 

 IJL = tan" 1 cot I cot x. Then we have 



cot X cot I 4- cos n L = tan + tan u. = sin * *., and 



COS 5T COS ft 



v a . cot I cosec 2 X cos ft cos ?r cosec + ,. 



CASE II. When cot* cot I = 1 ; that is, when the numerical va- 

 lues of x and I are complementary. 



Here + 1 +cos nL = 2 cos 2 



2 



1 -f cos nL = 2 sin' -" 



2 



And therefore the corresponding values of v are, 



i a TlL ut 



v = % a sec X cosec X sec 2 -rr 



' Xi 



o L 

 v = a sec A. cosec X cosec 2 . 



CASE III. When cot \ cot I ^ 1 ; we have, putting r = cot" 1 cot * 

 cot I, 



cos > + cos 7i L = 2 cos !( + L) cos ( "L)> and therefore, 

 v = %a cot I cosec 2 X sec ^( + nL) sec ( nL). 



* For, in this case, cot I = ^ .: cot I cosec 8 *. = . 1 .=-^J - = secXcosecX. 

 cos A cosX smX* sin X cos A 



