592 Messrs. Cowley and Lev}' : Method of Analysis suitable 



curve is to be integrated. From the graph the integral 



curve, 1 ^p- can easily be obtained, the mean -ordinate 



method being most convenient. Thus at 0*05 the integral 

 will be 0*05 times the average height of the curve between 

 and O05, at 0*1 it will be the area up to 0*05 plus O05 times 

 the average ordinate between 0*05 and 0*1, and so on. These 

 integrations can rapidly be performed, and if it is necessary 

 for the sake of accuracy to integrate thoroughly a portion of 

 the curve where the ordinates are small, this portion can be 

 plotted to a larger scale and allowance made accordingly. 



The actual details in the calculation will be omitted, but 

 experience shows that the various steps in the process can 

 easily be carried through accurately by anyone not even 

 conversant with the calculus. We then rind 



0=/i(l)/."(l) -/,(!) /, ;, (1) 

 = J (13-28 -t-2-M(9 4 + 0-00127^ 8 )(0-1174 + 2-4xl0- 4 (9 4 



-1-5-54 xlO- 8 8 ) 

 - (1 + 0-585 4 +- 1-033 xl0- 3 6> 8 +2*656 xlO" 7 xlO w ) 



x(l+0-601(9 4 + 2-36xl0- 4 (9 8 f o-l+lO" 8 ^ 12 ), 



i. e. 1 - 0-375 0H 0-015 s - 0-000072 12 . . . = 0. 

 This gives 



from which q is at once derived, and it will be seen that the 

 term in 6 12 may be neglected, the series being so rapidly 

 convergent. 



As a check, the whirling sp^ed in the present instance can 

 be found by an alternative method applicable when the shaft 

 is composed only of portions of uniform section. The method 

 consists in supposing the shaft simply supported at the 

 positions where the section suddenly changes, as well as at 

 the ends, and then utilizing the conditions that the reactions 

 at these intermediate supports are zero. The portions between 

 two neighbouring supports are uniform, and therefore the 

 functions / 1? f 2 , etc. are directly integrable algebraically*". 

 This method gives 



(9 4 = 2-91 



as against # 4 = 3 by the previous method. 

 * Advisory Committee for Aeronautics Report, Q & M. 690. 



