108 Mr DAVIES on the Nature oftfie Hour-Lines 



Or changing rectangular into polar co-ordinates, r, t, it becomes (divid- 

 ing all by cos 0) 



r cos x + a cosec A sec 6 , sin * cos I + sin I tan 



or separating the variables, 



a sec A cosec A sec i 



,-,f, 3-, T 



v sin- A + tan 2 > sec A tan I cos n cos 



_j (sin A cot I -f- tan 6) sin I 

 /sin 2 A + tan 2 + 



This expression admits of some simplification. For with respect to the 

 radical which is involved doubly in the denominator, we may write it 

 thus : 



sin 2 A -f tan 2 0=sin 2 A + sec*0 1 sec 2 6 cos 2 A= (sec 0+ cos A) (sec cosA)=g*, 



which is adapted to logarithms ; and A is the same for the whole dial. 



Again, for the numerator of the expression under the function cos" 1 , we 

 may put 



= tan" 1 sin A cot I. 



Then we have 



(tan > -f tan 6) sin Z = sin > + sec sec sin /, 



which is adapted to logarithms ; and and sec sin /, are constant for the 

 same dial. 



Further, sec A tan I, and sec A cosec A, are constant for the same dial. 



Lastly, if we make 



cos-' { sec A tan I cos n cor-' "nT+Taec.gecldn^ K> 



we shall obtain for the equation of the hectemorial curves, 



. K 

 r = \ a. sec A cosec A sec sec* . 



Or, exhibiting the formula at one view in a calculable state, it will be, 



a sec A cosec A sec 



r _ 



f .14 smj-^secsecsin 



i (sec i+cos A) (sec 0- cos AH sec A tan I cos n cos- 1 77 -~ r . r . 



{(sec0-fcosA)(sec-cosA)}i 



