46 Prof. Challis on an Extension of the 



to obtain rules for finding maximum and minimum values of 

 given functions. And as h and k are small indefinite quantities 

 having no relation to each other, such also is the case with 

 respect to &x and 8y. The former substitution is supposed to 

 change the form as well as the value of the function in which it 

 is made ; the latter only changes the value of the given function. 

 This being understood, the maximum or minimum condition 

 requires that the two parts of which u is composed should sepa- 

 rately vanish. With respect to B, this is effected by the values 

 of 8y and Sx at the limits of the integration, or given conditions 

 to which those values are subjected. But since the other part 

 contains the indeterminate and independent quantities Sy and 

 8x, it can vanish only by making the coefficients of these quanti- 

 ties vanish. Hence we have the two equations 

 A = 0, Ap = 0. 



These are differential equations the integration of which fur- 

 nishes the function or functions by means of which the required 

 maximum or minimum is calculated. I now make a remark 

 which has an essential bearing on what follows. It has been 

 usual to argue that because the second equation is satisfied if 

 A = 0, the two equations are always equivalent to each other, and 

 that it suffices to take account of only one of them. This, however, 

 as I am about to show, is a false conclusion ; and to this error the 

 difficulty of solving the before-named problem may be traced. 

 The second differential equation is not of the same degree as the 

 first; and it might happen that one is immediately integrable 

 and not the other, — an analytical circumstance which of itself 

 would make a distinction between them. To illustrate this 

 point, let us first take an instance in which the two equations are 

 equivalent, and then proceed with the discussion of the problem 

 in hand, which will be found to be an instance of the contrary 

 case. 



Let it be required to find the line of given length which ter- 

 minates at two given points, and encloses, with the straight line 

 joining the points, a maximum area. By the usual process we 

 obtain for this case 



Adx = dx — d 

 Hence 



Apdx = dy —pd . .. 'T.:.. - = dy — 



s/l+p* \/l+/ 



Here each of the equations Adx = and Apdx = is at once in- 

 tegrable; and each gives for the equation of the line, 



(x + cf+(y + c')*=\\ 



