Calculus of Variations by a New Method. 219 



occasion to review the process adopted in the above-mentioned 

 communications, I first ascertained that it did not give, as I 

 supposed, an absolute maximum, and subsequently that even 

 where it agreed with the method commonly employed, it was 

 essentially faulty. By admitting the principle of discontinuity 

 (the legitimacy of which will presently come under considera- 

 tion), and by reasoning in other respects according to the 

 received rules of the Calculus of Variations, a solid of revolution 

 was obtained such that the generating line of its surface is com- 

 posed of the two ordinates drawn to the axis at the given points 

 and a connecting curve. The analysis showed that the form of 

 the curve is that described by the focus of an hyperbola rolling 

 on a straight line, and it was proved that at its points of 

 junction with the ordinates it must be continuous with them. 

 As the rules of the Calculus of Variations indicated no other 

 line capable of satisfying the data of the problem consistently 

 with the condition of a maximum, it seemed to be a legitimate 

 conclusion that this line gave an absolute maximum. If, how- 

 ever, the points be very near each other, and the given super- 

 ficies be of great extent, it may easily be shown that the thin 

 solid which the surface encloses under these circumstances would 

 be much less than that enclosed by an equal amount of surface 

 generated by the revolution of a segment of a circle the chord 

 of which is the line joining the points; whence it necessarily 

 follows that some fallacy is involved in the above conclusion. 

 Having been convinced by these considerations that the problem 

 remained unsolved, and not admitting that the Calculus of 

 Variations could be at fault, I commenced a new investigation, 

 and, after much ineffectual research, at length discovered a prin- 

 ciple of solution which, I think, will be admitted to be legiti- 

 mate and to give satisfactory results. This new solution I now 

 proceed to develope. 



Having made trial of an investigation conducted by polar 

 coordinates r and 6 } I obtained in the usual manner the differ- 

 ential equation 



r sin #(r + ?■") r' cos 6—3r sin sin 6 



(Jv d v 



r 1 being put for -^ and r" for -^. Now it appeared that this 



equation did not admit of a first integration till it was multi- 

 plied by the factor r cos 6 -f r' sin 6, after which an integral was 

 obtained coinciding with that which results by employing rec- 

 tangular coordinates and multiplying the equation by the factor 



dy 



~. In short, the two factors, and also the two processes, are 



