1879.] 



the Capillary Phenomena of Jets. 



85 



length of maximum instability was, I believe, first considered in a 

 communication to the Mathematical Society,* on the " Instability of 

 Jets." It appears that the value of q may be expressed in the form 



2=v/ C 3 ) ' F<>) (7) ' 



where, as before, T is the superficial tension, p the density, and F is 

 given by the following table : — 



&% 2 . j 



Y{Jca). 



Tc 2 a 2 . 



'F(ka). 



•05 



•1536 



•4 



•3382 



•1 



•2108 



•5 



•3432 



•2 i 



•2794 



•6 



•3344 



•3 



•3182 



•8 



•2701 







•9 



•2015 



The greatest value of F thus corresponds, not to a zero value of 

 Jc 2 a 2 , but approximately to k 2 a 2 = '4<858, or to \=4'508x2<x. Hence 

 the maximum instability occurs when the wave-length of disturbance 

 is about half as great again as that at which instability first com- 

 mences. 



Taking for water, in C.Gr.S. units, T = 81, p = l, we get for the case 

 of maximum instability, 



= -115 di .... (8), 



* 81 x -343 v J 



if d be the diameter of the cylinder. Thus, if d=\, q~ 1 = , 115; or 

 for a diameter of one centimetre the disturbance is multiplied 2" 7 

 times in about one-ninth of a second. If the disturbance be mul- 

 tiplied 1000 fold in time t, gt=3 log e 10=6'9, so that t='79dl For 

 example, if the diameter be one millimetre, the disturbance is mul- 

 tiplied 1000 fold in about one-fortieth of a second. In view of these 

 estimates the rapid disintegration of a fine jet of water will not cause 

 surprise. 



The relative importance of two harmonic disturbances depends upon 

 their initial magnitudes, and upon the rate at which they grow. When 

 the initial values are very small, the latter consideration is much the 

 more important ; for, if the disturbances be represented by a^e^, a.^* 1 , 



in which q-± exceeds q 2 , their ratio is J^e - ^ - ?^ ; and this ratio de- 



creases without limit with the time, whatever be the initial (finite) 

 ratio « 2 : « P If the initial disturbances are small enough, that one is 

 ultimately preponderant, for which the measure of instability is 



* " Math. Soc. Prcc," November, 1878. See also Appendix I. 



