Bessel- Clifford Function, and its applications. 525 



a 2 — a* . ' 

 Replace '—=— in (4) by h for a constant tension 



length h ; then for a length a between fixed ends, not 

 clamped, take y = b sin ?tm?cos cot. ma — ir, for a plane vibra- 

 tion. And then if making N beats per second, 



— = — =m ck £ + mrh. 

 g A, 



Change the sign of h for a thrust, and let //, increase 

 gradually up to m 2 ck 2 ; N decreases to zero, and the vibra- 

 tion becomes sluggish : finally the bar ceases to beat, and is 

 sprung permanently on Euler's law ; X has become negative. 



For an upright rod, deflected slightly by vibration, or 

 ■bodily rotation, the D.E. (4) will change into 



ie .d*u d 2 u u „ , rN 



e *d? + *<&^x =0 (5) 



With c = 0, the equation reduces to (2) § 2, but with the 

 sign of x changed, for a chain hanging vertically downward. 



If the rod is drooping slightly under gravity, as in § 5, 

 \~co ; and with 



dif d 2 t< 7 »d 2 p 



pmt ai = -dj» ci ^ + ^ =0 ' 



with the solution, by (4) § 5, p = bG-J Q ~r: 2 ), of which the 

 smallest root is about 0'88, say §, making the critical height 



£ = 2(cfc 2 p = (icd 2 )^ 

 for a circular rod of diameter d. 



18. Looking back, then, on the rival claims of the Bessel 

 and Clifford function for employment in mechanical questions, 

 it would appear that the Clifford function has the advantage 

 of simplicity in the expression of results dealing with a 

 linear extension or propagation, as not usually requiring an 

 extraneous factor, a power of &\ 



And in any emanation Brom a centre or axis, it is more 

 natural to take r 2 or spherical surface \irr 2 as a variable, and 

 not ?', as implying that a negative r could arise, to be inter- 

 preted in a formula. 



But with r* as variable, say »/iV 2 = 4a? J a negative cc Mould 

 imply imaginary /', and a new set of conditions, so that ,ror r a 

 is more appropriate as a variable. 



