138 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



This, taken between the limits r = p, and r — ; and 9 — ?r, and = 0, gives 

 the total intensity 



4&RI'cos 2 2i/ ,,P , 

 u — to / e dd 



Teos 2 2i r 



*b 2 J 



o 



/ z 2 V 



Now, from Equa. 3, p = z cos 6 + a (1 — ^ sin 2 ) , s being less than b, 



.-. . - «yy { • (i - 5 -•'»)' + • - • } ■» ■ 



Expressed in terms of elliptic functions, 



4& RI' cos 2 2i T _, , A J . ."1 . n 



a E 2 (0) + 2 sin + C . 



u = 



*6 2 

 4& EI' cos 2 2i 



^i 2 



aE,(f) . . • • (8). 



But by Art. 1, R cosec 2i:c::b:^, (R cosec 2i being = FB, fig. 7), 



E cosec 2% 



b = 



2c 

 E 



2c sin 2i ' 



and 



a = b sec 2i 



cos2z 



Substituting in Equation 8, we get 



8ck I sin 2% cos 2% -r, , N 

 m = E z (?r) , 



_ 4c£I'sin4i _ , \ /n . 



= E* Or) . . (9). 



An expression which may be put in the following form : — If a circle be described 

 with F as a centre (figs. 7, 8, or 9) and 26 as a diameter, and an ellipse with for 

 one of its foci and the same diameter as a major axis ; the circumference of the 

 circle will be to the circumference of the ellipse as the intensity at the axis to the 

 intensity at 0. 



When z = 0, then E £ (tt) = tt 



j Ack V sin 4i , 7 T/ . . . , in , 



and u = x. <x = 4ck I sn^ , . . (10), 



it 



which is the expression for the intensity at the axis, and shows that in the same 

 conical mirror it is constant at every point in the axis ; whereas, in conical mirrors 



