162 Mr. C. Chree on Rotating Elastic 



increasing with ??. Thus for rj = *25 this critical value of b/a 

 is approximately *4034, and the approximate values of a— x 

 answering to the values '6, *8, and 1 of b/a are respectively 



•0045 a, -0150 a, and -0339 a. 



When r} = -3 it is obvious from (92) that x = a when 

 b/a—1. Also the coefficient of a 2 — 6 2 in the right-hand side 

 of (92) is easily proved to be positive for this value of 97 ; 

 thus x > a for all other values of b/a, or the radial strain is 

 never a compression at any point of the major axis in any 

 form of elliptic disk, though in a circular disk it just vanishes 

 at the rim. 



As 7) approaches close to *5 the coefficient of a 2 — b 2 in (92) 

 may become negative, but it is easily shown that the term 

 containing 3^ — 1 is then always sufficiently great to keep the 

 right-hand side of (92) positive. Thus when rj > *3 we have 

 x > a for all values of b/a, or the radial strain is an extension 

 at every point of the major axis. 



§ 42. Along the minor axis the radial strain is the value 



of -^f— with x=0. When v is less than 

 dy 



^(^13 — 3), or *3028 approximately, 



the radial strain along the minor axis diminishes algebraically 

 as y increases for all values of b/a. For larger values of rj 

 the radial strain diminishes or increases algebraically as y 

 increases according as b/a is greater or less than a certain 

 value depending on rj. This critical value of b/a increases 

 from when 77 = -3028, and approaches 1 as 97 approaches *5. 

 For 97 = -3 the critical value is i^/VW^S, or -3891 approxi- 

 mately. For ba = l, with 77 = '5, the radial strain has every- 

 where a constant value. When 77 = the radial strain is for 

 all values of b/a an extension at every point of the minor axis 

 except 'the ends, where it vanishes. When b/a is very small 

 the radial strain is a compression throughout the whole minor 

 axis unless 77 be very small. When both b/a and 77 are very 

 small, we find for points in the minor axis the approximate 

 formula 



f - &(?-?-*) (94) 



So in this case the radial strain is a compression along the 

 whole minor axis when b/a < x^rj, but for greater values of 

 b/a it is an extension between y = and y — V& 2 — *;a 2 - 



