Mr $teven$an^9 Totally Reflecting Mirror. 1,51 



y-y'=tan^(:r'-0 

 from which, since y"—o, and a/, aj", are both negative, 



and since a;' ;s; c? cos ^ y = c? sin "^ and 



)) 



from which by substituting the value of -^ 



2 ^=22-85362,* 

 and then from the formulae 



r=4:m is/2, a = 2 m and 6= —4 m^ 

 wefindr=64-640; a = 22-854; 6= -45707 



It will now be observed that the values of /* found by the two 

 formulae differ by nearly 4*77 inches; but this will not produce 

 a very great aberration of the rays, seeing the arcs of the 

 circles are so small that they will not deviate greatly from 

 each other. To ascertain the amount of the error intro- 

 duced by using the approximate value for r, we must bear in 

 mind that the circles calculated by the two formulae have 

 a common tangent at B. For in the osculating circla to the 

 parabola at B, 



dx y 2 m 

 and the tangent to the other circle at the same point by the 

 construction, makes an angle of 45° with the axis;, and 

 therefore, since one lies wholly within the other, they will 

 separate from each other most widely at A. We shall, in 

 order to estimate the error introduced by using the for- 

 mulae (2), calculate the difference of their ordinates at A, and 

 also the difference of their inclinations to the axis at that 



* It may be proper to remind the reader that this preliminary calculation 

 is only required here in order to compare the formulaB by calculating the value 

 of r for a mirror of exactly the same dimensions in both cases. In applying 

 the second set of formulae, originally, we might at once assume a value for m^ 

 and then the calculation would be much shorter than by the other method. 



