600 REPORT— 1887. 



obtained as a function of x, and the second -variation, when we leave out all small 

 terms, becomes, for an infinitely small range of integration, 



or say ^[(AS;/"'" + 2B5/")5sC") + cSs^''^'')*:, and the equation Sy = becomes 



so that eliminating 8z^"^ from the integral we get as the result that U is a maxi- 

 mum or a minimum for a very short range of integration, according as 



a( ^Y 2B J^ ^ + C (-'^^ V 



is negative or positive. 



If in V there had been no higher fluxions than •' and ~, the above 



rt.f dx"~'^ 



expression need only be changed by writing for '-^ ■ and '^^- , where they appear 



explicitly, ^^ _.,^, and -y-,-,-^,3, and putting zero instead of both where they 



appear implicitly in A, B and C. If in u and v, the highest fluxions be j/f'' and 

 z'-'\ and y^i'\ z'-'i'> respectively, and r~p>s — q, then the determining expression 



becomes , ' . AVhere there are more variables and more equations of connection, 

 di/'y- 



some patience is required to determine which terms must be retained, but the general 



principle is exactly the same. 



The method of deriving from these criteria the additional critei-ia necessary,, 

 when the range of integration is not small, is fully discussed in the paper quoted, 

 article 10, at least for the problems coming under class I., and it is quite easy to 

 see that the discussion is perfectly general. Owing to the limited space available 

 for the Abstract, it is impossible to include any account of it. 



When the limits are not fixed, there is no difticulty in determining the criteria, 

 provided there is but one independent vai-iable. But in the case of multiple 

 integrals, the \ariability of the limits gives rise to a problem of an entirely new 

 character. When, as is certainly often the case, the solution of the problem is 

 obtained in a form containing arbitrary functions of known quantities, the problem 

 depends on one of the following type. To find the form of ^ ao that 



shall vanish independently of the form of 8\j/. In this expression f^ and /„ are 

 known functions, and of course 8.\|/^( f,) is the same frmction of /j as ^.■^{jf^) is 



2. Sume Notice of a new Computation of tlie Gaussian Constants. 

 By Professor J. C. Adams, F.n.S. 



3. On the Umhral Notation. By the Rev. Robkrt Haeley, M.A., F.B.S. 



The germs of the system of notation proposed in this paper will be found in 

 Sir James C'oclvle's paper on Ilyperdistributives, printed in the ' Philosophical 

 Magazine' for April ls72; but the author is alone responsible for the form in 

 which the subject is here presented. He has endeavoured to develop and extend 



