Integration of Differential Equations of the Second Order. 29 



points, the radical and dx change sign together, so that 



1+ ~- 2 Ydx has always the same sign, which we may assume 



to be positive. 



Exactly the same considerations apply, mutatis mutandis, if 

 the curve be referred to polar coordinates ; and in the case of a 

 closed curve the pole may be at any point either within or 

 without it. 



The foregoing Lemmas, which ought to be introduced into the 

 Elementary Treatises on the Integral Calculus, I supposed to 

 be new, until it was pointed out to me by Mr. Todhunter on my 

 submitting to him Lemma I., that in his ' History of the Cal- 

 culus of Variations ' (art. 139) an application of the method of 

 integrating along consecutive points of a curve is recorded as 

 having been made by M. Delaunay in a memoir published in 

 1843. It is evident from this account that M. Delaunay pro- 

 duces the method as a recent discovery made by himself; but 

 the principle of it is not stated by him as generally as it might 

 be, no mention being made of integrating along a chord, or with 

 respect to intersecting portions of different curves. 



My purpose also requires a few preliminary remarks to be 

 made respecting the principles of the Calculus of Variations. 

 The elementary treatises on this branch of analytics give a me- 

 thod of obtaining a differential equation on the integration of 

 which the solution of the proposed problem depends. But they 

 do not usually take into account that, as in all problems involving 

 two variables x and y it is abstractedly possible to treat the va- 

 riations Bx and By as independent, and to obtain two differen- 

 tial equations, one of these is just as much as the other entitled 

 to be applied in the solution of the problem. Similarly, if there 

 are three variables x, y, z, three equations, all applicable to the 

 problem, are obtainable. (This method of treating the Calculus 

 of Variations is given in Francceur's Mathematiques Pures, 

 torn. ii. 2nd ed. art. 885.) The possibility of obtaining more 

 than one differential equation is a significant analytical circum- 

 stance which cannot be overlooked without restricting the ap- 

 plications of the calculus. When there are only two variables x 

 and y (as is supposed to be the case in all the subsequent rea- 

 soning), it may be shown in the usual manner that 



%-^){N-@-(g) + & c.}=0, , 



the differential coefficients being put in brackets to indicate that 

 they are complete (see arts. 8 and 9 of the " Calculus of Varia- 

 tions" in Airy's 'Mathematical Tracts'). Hence, if the pro- 

 blem involves no relation between By and Bx } we have, putting 



