282 Prof. Challis's Theoretical Determination of 



long, I must here refer for the demonstration. The equation 

 alluded to is 



from which, by assumingy to be a function of r, we obtain 



J dr'' ^dr' ^ r'dr^^^J -^' * ' ^^'^ 



whence it is clear that ify=0, -^ also vanishes. 



The integral of (9.) is derived from (7.) by putting ^ for^ 

 and —e for e. Hence 



1 „ 'eV ^r^ 



and 



/ = !_,,.+ _ ^+&c. 



The least value of r corresponding to/=0, as given by this 

 last equation, is the radius of a cylindrical surface within which 

 the motion of the fluid Jilament under consideration is con- 

 tained. It may be remarked, that the second term of equation 

 (9.) must be very small compared to the others (excepting 

 wherey approaches to zero), in order that that equation may 

 be equivalent to a linear equation with constant coefficients. 

 By the omission of the second term, equation (9.) becomes 

 identical with (6.). Hence the least value of r corresponding 

 toy=0 is very nearly the same as derived from either equa- 

 tion. Let I = this least value. Then the value of ^ is ob- 

 tained by finding the least root of the equation, 



Hence eP is a numerical quantity which may be calculated. 

 Let el^ = q. Then 



_ = ^= ^suppose. 



Hence k is constant for all vibrations if the ratio ^be constant. 



Now it may be thus argued that X and I have to each other a 

 constant ratio. These quantities must be related in some way, 

 otherwise the motion is not defined. Let F(A, /, o-) = express 



