for Differential Equations of Mathematical Pliysics. 591 

 where 



m . p, c r^ + ef ,, P | f ■*. fu ) v, r -? + . . . 



Jo Jo -tt Jo Jo -"-Jo Jo Jo Jo ^ 



mrf 1 "(i)=l+p( l d*{'rdmf'd*("Z£+ (21) 



Jo Jo ^o Jo ** 



( l J (' x !„ fee r*x 

 dm\ ™ dx\ /***+ (22) 

 -o Jo -ti Jo Jo 



^"(lj^pi \lx f %*<** + (23) 



Jo Jo 



For purposes of evaluation of the coefficients of the powers 

 of (9* in each of the above, it is to be noted that the first term 

 on the right of (20) will.be automatically.evaluated in deter- 

 mining the second term on the right of (21). which again will 

 be evaluated in calculating the second term on the right of 

 (20), and so on. The labour, therefore, in determining the 

 coefficients in the above four series is really only half what 

 it would appear to be at first sight. The geometrical outline 

 of the shaft, and the loading, are of course introduced through 

 the functions R and //,, and since all the integrals can be 

 rapidly evaluated by means of a planimeter, or otherwise, as 

 will shortly be seen, the most general case will be almost 

 as simple as any other by this method. As an illustration, 

 however, an example will be taken which is capable of solution 

 by another method in order that a comparison may be made. 



Example. — Consider a. homogeneous circular shaft simply 

 supported at the ends and built of three portions, the middle 

 portion being of diameter d and length Z/2, and the two ends 

 adjoining the supports of diameter d/4 and length Z/4. It is 

 required to find the first whirling speed of this shaft. 



The mid-section or the section of reference has a diameter d, 

 and w and I can easily be evaluated from this dimension. 

 jju will be 1/16 over the two end portions and unity over the 

 middle piece, while R will be 1/256 and unity respectively. 

 The series f\{x) and fi'(x) can be worked out simultaneously 

 as well as the two series /^(.r) and f%'(x), as can be seen from the 

 expansions. The steps in the calculation are evident. For 

 example, in evaluating fi(x) and//^,/'), w/R is first obtained 

 for various values of x. Each figure is then divided by the 

 maximum occurring in the column for convenience in 

 plotting, a method that may be used throughout whenever a 



