598 REPORT— 1887. 



of the length of the storm — instead of round a short nearly vertical axis, as in 

 cyclones. 



The outline is given of a proposed scheme for the systematic observation of 

 thunderstorms in England, by which it is hoped that the mechanical nature of the 

 circulation of the air in every kind of thunderstorm may be discovered. It is also 

 shown that if that particular kind of thunderstorm which is not associated with any 

 distortion of isobaric lines can be worked out, a kind of rain could then be success- 

 fully forecast which is now ver}- rarely announced. Forecasts now have to depend 

 almost exclusively on synoptic charts of isobaric lines. When these fail, they fail ; 

 but it is hoped that observations on the form and motion of clouds may be found to 

 indicate the approach of rain when the barometer shows nothing. 



Mathematical Sub- Section. 



1. On the Criteria for Discriminating between Maxima and Minima Solu- 

 tions in the Calculus of Variations. By E. P. Cclverwell, M.A. 



The paper explained the mode of finding the criteria for all known classes of 

 problems, provided the limits be fixed, and when the limits are not fixed, the nature 

 of problem to be solved was indicated. There are four classes of problems. 



I. TomakeU-[[ . . . [ / C*!; •% • • • ■t'm,!/v!/.,, ■ ■ ■ i/u)d.r, d.r^ . . . d.r,„ a 



maximum, / representing a function including differential coefficients of the y's by 

 the x's. 



II. Suppose V, W, &c., to represent integrals of the same character as U, we 

 may be required to make F(U, V, W . . . ) a maximum where F represents a 

 function of known form, and they,, y.,, . . . y,, occurrinfr in U, V, W, . . . are the 

 quantities whose form is to be determined. 



III. It may be requu'ed to make U a maximum subject to the condition 

 V = constant ; or, more generally, a similar restriction may be applied to problem II., 

 modifying it as tliis problem modifies I. 



IV. It may be required to make U a maximum when the variables .i\, 

 .tj . . . .i„„ y,, y^, . . . y„ are connected by one or more algebraic or differential 

 equations, or this restriction may be introduced in problems II. and III. 



In all these cases the criteria consist of two parts : 1st. There is a condition or 

 set of conditions which must be satisfied for every possible set of values of the inde- 

 pendent variables within the limits of integration. 2nd. The limits of integration 

 must satisfy certain conditions. 



The first set of conditions is obtained, without any algebraic transformations, 

 by taking an infinitely small range of integi-ation and showing that, when the 

 limits are fixed, only the ' highest differential coefficients ' of the variations need be 

 retained, both in the integrals and in the equations of connection. A full account 

 of the method of comparing the orders of magnitude of the variations may be 

 found in the ' Transactions of the Royal Society,' vol. 178, p. 95 ; but for the 

 simple case in which there is but one independent variable, all we need to do is to 

 point out that, because 



-^-i must be infinitely small compared to ■ ^ when the limiting value of . — •' 



is zero, and the range of integration is infinitely small. Now, when the 'limits are 

 fixed' the limiting values of all the valuations appearing outside the sign of integration 

 in the most reduced form of the first variation (the form which enables us to deter- 

 mine the value of y giving the maximum) must be zero, and therefore they must 

 all be infinitely small compared with the variation of the highest differential coeffi- 

 dent appearing in the function to be iutegrated. Hence the value of the second 



