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III. On the Application of a new Integration of Differential Equa- 

 tions of the Second Order to some unsolved Problems in the Cal- 

 culus of Variations. By the Bev. Professor Challis, M.A., 

 F.R.S., F.R.A.S* 



BEFORE stating the principle of the proposed new integra- 

 tion, and applying it in an example, it will be proper to 

 direct attention to the following two Lemmas : — 



I. The formula \ ydx may express the area of any curve between 

 assigned limits of the abscissas, whatever may be the origin and 

 direction of the rectangular coordinates, and whether y has one 

 or more values for a given value of x, provided always the inte- 

 gration be taken along consecutive points of the curve. 



II. Under precisely the same conditions 1(1 -j- —-^Ydx may 



express the length of any curve between assigned limits of the 

 abscissas, the radical being always taken with a positive sign. 



To exemplify the first Lemma, suppose the curve to be a 

 circle situated within the quarter for which the signs of x and y 

 are both positive. Then if we begin the integration with the 

 less of the two ordinates corresponding to a given abscissa, and 

 proceed along consecutive points of the curve in the direction of 

 the movement of the hands of a watch till we arrive at the ex- 

 tremity of the other ordinate to the same abscissa, the integra- 

 tion between these limits will be the segment of the circle cut 

 off by this ordinate. For the integration from the given ab- 

 scissa to the minimum abscissa is negative, because through 

 that interval dx is negative; and the integration from the 

 minimum abscissa to the given abscissa is positive, because in 

 the return direction dx is positive. Also the positive area is 

 made up of the negative area and the above-mentioned segment 

 of the circle, so that the total integration gives the area of the 

 segment. 



The same principle applies, "however complicated the curve 

 may be ; only to effect the required integrations it is necessary to 

 obtain all the real values of y corresponding to any value of x, 

 either exactly or by approximation, as explicit functions of x. 



If it be required to calculate the area enclosed by a portion of 

 the curve having its extremities at given points and the chord 

 joining the points, the same rule applies, if, after integrating 

 from one point to the other, the integration be continued along 

 the chord to the first point. 



With respect to the second Lemma, it is only necessary to 

 remark that when the integration is taken along consecutive 



* Communicated bv the Author. 



