for Differential Equations of Mathematical Physics. 585 



methods demanding the use of a slide rule, a multiplying 

 machine, or planimeter. Graphical processes depending 

 on differentiation ought to be avoided, as experience shows 

 that without a considerable degree of labour inaccuracy 

 is unavoidable. In the second place, if the expansion is 

 in series the convergence should be as rapid as possible 

 and the rate of convergence should be more or less evident 

 at each stage. Of the common methods of expansion, 

 the Fourier's series type is perhaps the most suitable so 

 far as these considerations are concerned. An ordinary 

 povver series, on the other hand, may or may not be 

 rapidly convergent over the whole range according to the 

 nature of the problem considered. The crux of the matter 

 is reached from a third consideration, which requires that 

 the form of expansion should be sufficiently expressive to 

 allow of further analysis of the solution to determine the 

 properties of, say, the class of problem under consideration 

 and the particular member desired. Consider, for example, 

 the problem where a series of struts whose law of cross- 

 section is given for the whole class, say I = I R, where I 

 and I are the moments of inertia at any section x and at a 

 standard section respectively, and R is a non-dimensional 

 function of x specifying the law of variation in I along the 

 length. 



Let F be an end-thrust acting longitudinally along the 

 strut and I the length of the strut. Let the problem be, 

 to determine the particular member (that is to say, the 

 value of I) of this class which when under given eccen- 

 tricity will give a deflexion a, at the middle, the strut being 

 simply supported at the ends. The differential equation for 

 the flexure of the strut is 



when the origin is taken at one end. 



This of course holds for all values of EI 0? F, and /, 

 and therefore represents apparently at first sight a four-fold 

 infinity of problems ; but it is easily shown that this is 

 not the case, but rather that the passage from one member 

 of the class to another is brought about by the variation 

 in value of one expression involving all these quantities, 

 so that in reality there will exist merely a one-fold infinity of 

 members in the class. Writing x = lx', j/ — l}/' and inserting 

 this in the differential equation we find, dropping dashes, 



F/2 dx*+ { K °' 



where = ^ and R is a function of x. 

 Jill,. 



