«7i the usual process for forming a Plane Surface, 111 

 second by p'={q")^, 



y.A„+y./.B„=E.( ? ')" +iw . 



p<.A n+ p'. P ".-B„=-E.(q"r i+ *> 

 and subtracting the upper from the lower, 



(p'-p"). A. =E . { - (q')" + i +fi + (q"f +i ^} i 

 or 



-Vf" ^^ . sin (« + w) . A n 



= -2E . (^ + « + * • n/^1 . sin (w+l+/3 . 37+3^), 

 from which 



a/2 



i .P 



A n =+E.-_4^.(lV + 6 + i.sin( w + i+^.3« + 37r\. 

 sin (a + 7r) \ 8 / V ■ a J 



The complicated constant E _ may be expressed as a 



sin (a + 7r) 



single constant F, and then we have 



. - v n 1 /3 



A„ = + F . (g )2 + 6 + 2, sin (»+ 3 -f/3 • 3« + 3tt), 



B n = -F . (y)* +i " . sin (n~+~P • 3a + 37r). 

 The constants F and /3 are to be determined by making 



A = +F . ( j)* + *. sin (|+/3 . 3^+3tt), 



B = -F . (j)-2. s in OS . 3a + 37r). 



It has been found by actual substitution that the values found 

 for A ra and B n satisfy the original equations. I consider this 

 proof of correctness to be necessary when real values are in- 

 ferred from a process conducted by means of imaginary quan- 

 tities. 



It is worthy of remark that the expression for the amount of 

 prominence consists, in each case, of the product of two terms 

 which vary with the number of operations. The second term is 

 periodical, showing that the prominence may even change sign. 

 But the first shows a rapid decrease in geometrical progression : 

 one operation makes a reduction in the proportion of 14 : 5 

 nearly ; two operations, in the proportion of 8 : 1 ; four operations, 

 in the proportion of 64 : 1, &c. The rapid approach to a truly 

 plane surface is thus explained. 



Royal Observatory, Greenwich, 

 July 11, 1871. 



