MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
97 
Jacobi did not himself give a detailed account of his process, but said that “ the 
analysis just indicated requires a good knowledge of the Integral Calculus.” Various 
demonstrations were subsequently given by different mathematicians. That of 
Delaunay has been adopted by Jellett and other English writers. 
In 1852 Mainardi devised a method somewhat similar to that of Brunacci, but 
he endeavoured to remedy the omission in the latter by showing how to determine 
the coefficients used in the transformation, the equations for determining these 
coefficients being supplied by Jacobi’s reasoning, But in this he was not successful? 
even in some of the simple cases which he discussed, and in the more complicated 
cases the equations appear to be quite unmanageable. 
In 1853 Eisenlohr extended Jacobi’s method to double integrals. In 1854 a 
memoir by Spitzer was published which seems to have been more complete than 
Mainardi’s ; but the most important advances were made a few years later when the 
Theory of Determinants was applied by Hesse and by Clebsch to simplify and 
extend Jacobi’s methods. 
From the foregoing sketch it will be seen that as early as 1806 the criteria had 
been correctly given and simply proved, with the exception of one point, namely, 
that there was no means of ascertaining for what range of integration the criteria 
ceased to be sufficient. Jacobi endeavoured, by the help of a complicated analysis, 
to remedy this defect; and, although all the efforts of later mathematicians have 
been directed to the extending or simplifying of Jacobis method, the analysis is still 
very complicated, and requires an intimate acquaintance with other branches of 
mathematics. 
All these methods, however, are open to the objection stated in Art. 5, and, further¬ 
more, it appears to me, for the reasons briefly indicated in Art, 12, that, although the 
results arrived at by these mathematicians are undoubtedly correct, it would be 
impossible to give a strict proof of them by any method based on transformations. 
However this may be, I cannot find that anyone has yet given a proof of them. 
Jacobi merely states the limits within which the criteria hold. 
The chief object of the present paper is to show that a rigorous discussion of the 
discriminating conditions can be given without introducing any analytical trans¬ 
formations whatever, the results being obtained by reasoning from the fundamental 
conceptions of the Calculus of Variations. In the first Part of the paper will, how¬ 
ever, be found an analytical method leading to Jacobi’s transformation, but free from 
any serious difficulty. It is inserted chiefly on account of the historic interest of the 
problem. I had extended this method to obtain the criteria for the case of any 
integral whatever before I was aware that the results were not altogether new. It 
was after finding the limits up to which the criteria were sufficient that I was led to 
the general method given in Part II. 
For convenience, a summary of contents is given below; the numbers refer to the 
MDCCCLXXXVII.-A. 
O 
