322 Mr. Cankrien on the Problems of 



times invariable, and sometimes variable : the problem of the 

 brachystochron is an instance of one or other of these two 

 classes of problems, accordingly as we investigate the line of 

 quickest descent from one given point to another given point, 

 or from one given curve to another given curve. . 



If then Y=f(x, y 9 p, q, r, &c.) where p = -£. ; q -.JL ;r 



= -—■; &c. ; U = « +fdxY', « being an arbitrary constant; 



and U/— U ;/ be the value of U between the limits x t y t \ x u y tl : 

 our object is to find the relation between x and j/, which ren- 

 ders U, — U y/ a maximum or minimum ; either when x t y ft x u y u 

 are invariable, or when they are variable and connected by 

 given equations y t = <J> (*,), y u = ty (x„). 



Now it is to be observed, that if we can determine the form 

 of the function F (x) in the equation y — F (#), which renders 

 U,— \Jj! a maximum or minimum, when the limits are invari- 

 able ; we can also find it by the same process, when the limits 

 are not assigned, but when equations only are given connecting 

 them: for we may suppose that the symbols x^^ x n y jft which 

 represented the limits which are supposed invariable in that 

 process, now represent those values of the variables in the 

 equations y / = <p (.r,), y n = rj/ (x^) which must be taken as the 

 limits of the integral in order that U / — U /; shall be a maxi- 

 mum or minimum. The only difference in the results we 

 shall obtain in the two cases will be this : when the limits are 

 assigned, we can substitute their given values for the symbols 

 Xj Vm x n y lP and thus find determined values for one or more 

 of the constants in the equation y = F (x) ; whereas when we 

 have not the values of the limits assigned, those constants 

 which may be expressed in terms of these symbols will con- 

 tinue arbitrary, unless we have some method of determining 

 the values of the limits. We will then for the present con- 

 sider the limits of the integral not to change. 



We supposed U = u +fdxY when the function of x re- 

 presented by y is involved in V: if y = F (x) then U / — \J n is 

 the maximum or minimum value oifdx V when taken between 

 the limits x / y / and x u y u . Let u be any function of x which 

 vanishes when x t or x n is substituted in it for x ; and letk be 

 a very small constant quantity ; also let W be the value of a + 

 fdxY when y + k u is substituted in V for y. Then W ; — W 7/ 

 is greater or less than \J / —JJ // according as U ; — U rj is a 

 minimum or a maximum, whatever be the function u. Now, 



when y + h u is substituted for y in p f it becomes p + k — = 



p + k u' ; g becomes q + k u" ; r becomes r + h u'"\ &c. ; and 

 therefore since W is the value of U when y + ku is substi- 

 tuted 



