378 Rev. B. Bronwin on the Itectification of tlie Ellipse. 



the rays to have crossed, and the point which the second Ex- 

 periment determines to be anterior to their crossino; ; and as 

 these extreme points approximate closely to each other, as in 

 the fii'st case the focus is within, and in the second without 

 the axis, and as the retina is situate between them, — I consider 

 my proposition established ; viz. " that the focus of the eye is 

 upon the retina." 



J. R. RUMBALL. 



LXI. On the Rectification of the Ellipse. By the Rev. Brice 

 Bronwin*. 



T^HE well known differential expression of the length of the 



arc \s dx V 'i- — (^ X sin^ x =. d L^ / _i"Jlli-|. 2 cos 2 j 



2 y e^ 



=. -- dx \/ p -\- '2. cos z by substitution. 



Assume \/ p + 2 cos s = A + A cos z + A cos 2 z + &c. 

 Difference both sides for z, divide by d z, multiply by 

 jo 4- 2 cos z : the result will be the same as if both sides 

 had been multiplied by sin z, being the expression of sin z 

 -v/ ^ + 2 cos ~. By equating thei'efore the coefficients of the 

 like terms in both the developed expressions of this quantity, 



52 1 73 211 



we get -^ A + p A — A = 0, -2~ A + 2^ A + "^ A = 0, 



94 332 _ 2 3 



-yA + 3pA+ -^A = 0. The quantities A, A, &c. are 



1 

 therefore known when A and A are known. 



1 2 



Resuming therefore v' p + 2 cos z= A + A cos z + A cos 2z 

 4- &c.; difference both sides for p, and then multiply by 



p + 2 cos z ; the result will be the expression of :§ V p+ 2 cos z. 

 Equating therefore the coefficients of like terms in both ex- 



1 



pressions of that quantity, we obtain — h p —— A= 0^ 



21, 2 

 \- V h 2 7, A = 0,&c. Eliminating — — from 



dp ^ dp dp 2 ' o dp 



the second of these by means of the first of the former equa- 

 1 1 



tions, we have p -, \- 4 w" A = 0. Eliminating -;— 



' * dp dp 2 <^ d]t 



* CDinmunicated by tlie Author. 



from 



