H3-115] Rotating Cylinders 113 



Writing e for &', and calculating the curve in the two-dimensional problem 

 as far as e 5 , we find for the equation to its surface expressed in polar coordi- 

 nates 



r* = (1 + -139e 2 + -023e 4 + ...) - '2ller cos 

 + (-8 - -138e 2 - -069e 4 + . . .) r 2 cos 20 

 + ('063e - '0064e 3 - -0031e 5 ) r 3 cos 30 

 + ('013e 2 + 'OOOSe 4 + . . .) r 4 cos 40 4- ('00360 3 + -00093e 5 + . . .) r 5 cos 50 



+ (-0001e 6 +...)r 8 cos80 + ....................................... (338), 



while the equation determining o> 2 is 



............... (339). 



115. The intersections of the curve with its longest axis are given by 



<I> (r, e) = 0, 



where 



(-2 + 138e 2 + -069e 4 + . . .) r 2 + (-063e - '0064e 3 - '0031e 5 + . . .) r 3 

 -0008e 4 + . . .) r 4 + (-0036e 3 + -00093e 5 + . . .) r 5 

 ...)r 6 + C00043e 5 + ...) r f+... . .................... (340). 



In this equation only a few terms are written down of the doubly infinite 

 series which represents the true value of <. For small values of r and e 

 these terms will give the value of < with considerable accuracy, but for larger 

 values the approximation may fail. We require to determine over what region 

 of values of r and e the terms actually written down will give a good approxi- 

 mation to the whole. 



The coefficient of each power of r is an infinite series, of which terms up 

 to r 5 have been calculated. The approximation provided by these terms is 

 seen to be tolerably good so long as e< 1, but fails when e exceeds a unit 

 value. 



When some definite value less than unity has been assigned to e, the value 

 of <1> will be given by an infinite series of powers of r of which the first seven 

 only are known. For small values of r these first seven terms will give a good 

 approximation ; for higher values of r the approximation will be poor, while 

 for still greater values the series will become divergent, and the first few 

 terms will give no approximation at all. Inspection of equation (340) shews 

 that the approximation will be tolerably good so long as r 2 < 1/e 2 . 



j. c. 8 



