1879.] 



the Capillary Phenomena of Jets. 



81 



dimensions of P, a force divided by an area, are 1 in mass, —1 in 

 length, and — 2 in time. Assume 



X-ocT^A'P*; 



then by the method of dimensions we have the following relations 

 among the exponents — 



x +y + u =% —3 y +2z—u=(>, -2x—2u=0, 



whence u=—x, y = 0, ,8=1(1— x). 



Thus XocT*A*-**P-* oc A} 



VPAV ' 



The exponent x is undetermined ; and since any number of terms 

 with different values of x may occur together, all that we can infer is 

 that X is of the form 



where / is an arbitrary function, or if we prefer it 



\=T- 4 P^A 



where P is equally arbitrary. Thus for a given liquid and shape of 

 orifice, there is complete dynamical similarity if the pressure be taken 

 inversely proportional to the linear dimension, and this whether the 

 deviation from the circular form be great or small. 



In the case of water Quincke found T = 81 on the O.G.S. system 

 of units. On the same system p = l ; and thus we get for the frequency 

 of the gravest vibration (n=2), 



i^=3-51a -l =8-28A-' .... (3). 



For a sectional area of one square centimetre, there are thus 8*28 

 vibrations per second. To obtain the pitch of middle C (c' = 256) 

 we should require a diameter 



2a=(— Y=-115, 

 V2-56/ 



or rather more than a millimetre. 

 Por the general value of n, we have 



£=l-43a-V(» s -»0 = 3'38 A-V(* s -») • • W- 



2tt 



If h be the head of water to which the velocity of the jet is due 



3-38y0 3 -^) 



VOL. XXIX. G 



(5). 



