52 Prof. Challis on an Extension of the 



and y 2 having a common abscissa, the expression for the seg- 

 mental area ^{y'—y u )d% will be 



jo x and jo 2 being the values of jo for the ordinates y 1 and ?/ 2 . In 

 the March Number I have shown by simple geometrical rea- 

 soning, which need not be repeated here, that, on the supposition 

 that the line is a circle, this expression gives the area of the cir- 

 cular segment cut off by the difference of the limiting ordinates. 

 Hence it may be inferred that the form of the curve which 

 satisfies the condition of a maximum can be no other than that 

 of the circle, and that the position of its centre is at disposal for 

 fulfilling required conditions. 



I find that Mr. Todhunter, in the communication before 

 referred to, has objected to this solution also that it does not 

 satisfy the equation A = 0. This objection has apparently been 

 made without taking into consideration that the function A, 

 because it contains a radical, has two values, and that the prin- 

 ciples of analysis demand that both should be taken into account. 



. ds 



Since the radical has arisen from substituting vl -j-^> 2 for -r-> 



it implies that there are two ordinates to the same abscissa, and 

 that the two sets of ordinates belong to parts of the curve which 

 have their convexities turned opposite ways relatively to the axis 



of x. In that case — , has the same sie;n for both sets. 



Hence, using dashes to distinguish between them, we have the 

 two equations 



2y ! dx=\ds'-'d.-^ /,pl 



tyd»=—Xd#-d 



XyY 



and by subtraction, 



This last equation, inasmuch as it takes account of both values 

 of A, is the one which the form of the curve is required to satisfy. 

 To draw any inference from one value of A and exclude the other 

 would be nothing short of error. By integrating, and determi- 

 ning the arbitrary constant so that the area J [ij —y n )dcc, and the 

 arcs a 7 and s" commence where p'z=p"=z infinity, and y ! =y n = y Q , 



