Calculus of Variations by a New Method. 227 



to y = b, and doubling the results, there will be obtained for the 

 length of the upper part of the curve lying between the extreme 



ordinates 2ira — 2a cos -1 -, and for the length of the lower part 



1 e 



2a cos -1 -. 

 e 



The differential expression for calculating the area is 



from which by integration may be obtained 



This integral, taken from y=*a(e — 1) to y=a(e+l), is 7nz 2 , 

 which, according to what has been already said, is the area of the 

 left-hand segment cut off by the maximum ordinate. Hence the 

 whole area of the oval is 27ra 2 . 



By integrating from y = a(e — 1) to y = b, and from y = b to 

 y = a(e+l), the double areas will be found to be respectively 



— 7TA 2 + 2ab + 2a 2 sin- x -, 

 e 



and 



37ra 2 ~2a&-2« 2 sin- 1 i. 

 e 



The first of these is equal to the area bounded by the extreme 

 ordinates, the included abscissa, and the lower part of the curve, 

 and is negative, because dy and dx have different signs from 

 y = a(e — l) to y = b. The other area is that bounded by the ex- 

 treme ordinates, the included abscissa, and the upper part of the 

 curve, and is evidently positive. It is not possible from these 

 results to calculate the areas of the two parts into which the oval 

 is divided by the straight line joining the extremities of the 

 limiting ordinates, because the length of this line can only be 

 known by obtaining y as an explicit function of x. 



Cambridge, February 20, 1866. 



