Calculus of Variations by a New Method. 221 



and it is particularly to be observed that the arbitrary constant 

 c may be different for the two equations. Thus we have been 

 led by rigid deduction to the equations of three lines, — one a 

 straight line parallel to, and at an arbitrary distance from, the 

 axis of v, which is the axis of revolution ; and the two others 

 circles of radius \ having their centres at arbitrary positions on 

 that axis. 



Before proceeding to the next step of the reasoning, it will be 

 proper to direct attention to an analytical principle which I have 

 discussed in the September Number of 1862. It is there re- 

 marked that the processes of the calculus of variations generally, 

 answer a proposed question by furnishing a differential equation 

 between the variables, and that such equation requires to be 

 afterwards treated according to appropriate rules. Now it is 

 known from the theory of differential equations, that generally 

 when the degree of the equation is higher than the first, the 

 process of integration conducts to more than one solution. When 

 the differential equation furnished by the calculus of variations 

 is of this class, there appears to be no other legitimate course 

 than that of employing the several solutions conjointly in order 

 to satisfy the conditions of the problem, supposing no criterion 

 to exist by which certain of the solutions might be shown to be 

 inapplicable. In the paper just referred to, an instance is given 

 of such reasoning by the solution of the problem of the brachis- 

 tochronous course of a ship from one given position to another, 

 the analysis conducting to the equations of 'two straight lines 

 passing through the positions, and inclined at supplementary 

 angles to the straight line joining them. The ship's course is 

 accordingly partly on one of the straight lines, and partly on the 

 other. 



The same principle being applied in the problem before us, 

 we have to satisfy the condition of drawing from one of the given 

 points to the other a composite line, consisting of parts of two 

 equal circles having their centres on the straight line joining the 

 points, and of a straight line parallel to this line. Clearly this 

 may be done by making the circles pass each through a point, 

 with their concavities turned towards each other, and connecting 

 them by the straight line. But our argument would still fail 

 unless the analysis indicated the mode of junction of the straight 

 line with the circles. To determine this, recourse must be had 

 to the part of the variation freed from the sign of integration ; 

 which, if the integral be taken from y = y to y=y v gives, by 

 being equated to zero, the following equation : 



^i^fyi _ M)?/o&/o _ Q 



