130 Mr. H. A. Webb and Dr. J. R. Airey on the 



design, including probably the over-all dimensions, are 

 already settled by various considerations with which we 

 are not now concerned. But we are allowed some latitude 

 in detail design, which we are to use to avoid elastic failure, 

 or to avoid, or to secure, resonance, as the case may be. We 

 want therefore to be able to calculate, roughly but quickly, 

 the effect on the crippling load, or the period, of various 

 possible alterations. We want in fact to make several trials — 

 the more the better — and choose the one we like best. Finally, 

 when the design is complete, we wish to check it carefully by 

 a more accurate calculation. 



The functions f(x) and (f>(x) in equation (1) are to be 

 -considered, for a tentative design, to be defined by their 

 graphs, which must be represented, for the range of values 

 of x required, by empirical formulas, the closeness of the 

 representation giving some idea of the accuracy to be 

 expected in the solution. These empirical formula? should 

 be of the simplest type, e. g. polynomials, or the ratios of 

 linear or quadratic functions of #, otherwise time is wasted 

 in constructing them. What is required therefore is a list 

 of suitable equations of the type (1) that are soluble in terms 

 of tabulated functions. The two important characteristics 

 are that/(^) and <f>(x) should be of a simple type, and that 

 they should contain several arbitrary constants; we can then 

 hope to make them fit our graphs fairly well without much 

 trouble. 



When/(V) and <£(#) are constants, the solution in terms of 

 circular and exponential functions is well-known. A useful 

 list of equations soluble by Bessel functions, with appropriate 

 tables, has been given by Jahnke and Emde*. It is the 

 object of this paper to show the value, from this point of 

 view, of the confluent hypergeometric function, tables and 

 graphs of which are given in § 4. For quick work graphs 

 are more convenient than tables. A list of differential 

 equations likely to be useful to designers, and soluble by 

 means of these tables and graphs, is given in § 3. Some 

 properties of the functions that were used in constructing 

 the tables, and would be useful in extending them, are given 

 in § 2. 



It may perhaps be argued that few engineers have the 

 mathematical ability for such scientific methods of design. 

 But it should be remembered that many engineers acquire at 

 their technical college or university a high degree of mathe- 

 matical skill; and if they lose it afterwards, it is because 



* Funktionentafeln, Teubner, 1909. 



