XCIV. NEW SOUTH WALES LIGHTHOUSES. 



.'. r 



dO 

 dr 



cos 0-fx 

 sin 



.'■ r sin 



O.dO-- 



= dr. (cos 6~ fj) 





dr 

 r 



cos - fX 



/» sin 



integrating / A 



.dd-- 



pdr 

 ~J r 







=log r + log C 



where is 



an arbitrary constant, 



and from (2) 







- log (fji - cos 



!<?) = 



= log (r C) 



.-. r 





1 





fx, — cos 



and r 





C 



(2) 



(3) 

 [X — cos v ' 



which is the equation to the curve, C being an arbitrary 

 constant depending on the radius of the lantern, and the thickness 

 of glass at centre. 



If FQ = a 



i.e , r = a when 6=o 



we have from (3) 



C 



a = . 



/x- i 



i.e., C =a (fji-l) 



and the equation becomes 



r= a -^^l (4) 



fi - cos 6 v ; 



When investigating the nature of the curvature of the 

 different prisms used in the dioptric apparatus of our lighthouses, 

 it seemed to the Author that as in none of the prisms used were 

 there less than two refractions, if the number of refractions 

 could be reduced, a saving in light could be effected. It is 

 possible to reduce the number of refractions to one by making 

 the inner surface of the lenses spherical in form, the radiant 

 point being the centre from which the sphere is described. The 

 issuing rays of light will then strike the inner surface normally, 

 and consequently there will be no refraction at this surface. 



