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XXIII. 



ON THE COI^DITIONS FOR MAXIMUM AND MINIMUM 

 SOLUTIONS IN THE CALCULUS OF VARIATIONS 

 WHEN CERTAIN FLUXIONS OF THE VARIABLES 

 HAYE FINITE AND ARBITRARY VARIATIONS. By 

 E. P. CULYERWELL, M.A., F.T.C.D. 



[Eead June 12, 1899.] 



Attention has been recently directed to this problem by the investiga- 

 tions of Weierstrass and Zermelo in Germany. The former gave the 

 condition for two dependent variables, yi and y^^ when they are con- 

 nected with the independent variable by an equation such that the 

 three really represent a plane curve, and when only the first differentials 

 appear in the function to be integrated. The latter extended the result 

 to the case where higher fluxions d'^yxjax"' and d''hj.,\dx'^ appeared, and 

 where these quantities alone might have finite variations, the same 

 equation of connexion holding between the variables. 



The investigations were very tedious, but the result was very 

 simple ; and in seeking for a proof dependent more or less on geometric 

 ideas, I extended the result, first to one independent and any number 

 of dependent variables, and, by a somewhat longer process, to the case 

 of multiple integrals, and found that I could also give the criterion 

 where any number of higher fluxions were permitted to take finite 

 variations. As I have since seen how to prove the result for all cases 

 of single or multiple integrals by a single and very short method, I 

 am not giving to the Academy the original rather long investigation 

 for multiple integrals, but that for single integrals is here presented. 



Since the variations may be finite and arbitrary, they must be 

 capable of sudden changes from one finite value to another, and the 

 very conception of such discontinuity involves the condition that the 

 integral taken along the discontinuous variation must be equal to the 

 sum of the integrals taken from one point of discontinuity to the next. 

 For if it were necessary to treat a sudden change of direction, for 

 instance, as a limiting case of a line changing its direction by turning 

 round a point, so that the angular point of discontinuity was regarded 

 as an indefinitely small circle, and as such might give rise to a finite 

 element in the integration, that would be to treat the variation as 

 continuous, though finite, and not as arbitrary and finite, 



B.I.A. PEOC, SEE, III., VOL. V. 2 E 



