348 MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS 



The determination of X, is easiest carried out in the following manner : 



= c \ *> T - &c 



In nearly all cases it will be sufficiently correct to take X,, = X" H . 



If there is only one vibration (3) present, the equation for the surface of the jet 



will accordingly be 



r = n n + b n cos (>t<> + f n . cos 2irzX H . 



If all the partial vibrations are taken in the same manner, we have, by the addition 

 of the results, the following equation for the surface of the jet 



/ = a a + 2 . &. cos (< + e B ) . cos(27T2/\ B ) (6). 



As an example, we will here calculate the vibration of a jet the original cross- 

 section of which is an ellipse with the axis 2a and 2b = 2 (a 8). 

 The polar equation of the ellipse is 



ab a 8 



r = 



v/a 3 sin 3 < + b 3 cos 3 <f> \/\x cos 2 

 where 



(7), 



-2 8 - 82 



^ 5- 



By expanding in series 



. 

 a cr 



, COS 2A^ -lj-4- 1 i-^V + Il 1 3-S :r3+JL iiJ-iA.--',,.* . 1M .l_- .?._ yS. \ 



I- OUb -i<p.^ a a-^^a 2.4 X ^39' 2.4.6 <1// ^16 2.4.6.8^ ^256 2.4.6.8.10'*- T ) 



H>04^^ ^.S~ . 1_-JL^53 , _7_ 1.8.5.74.15 1.8.5.7. 9 g , N 

 *9-U ' 2 . 4 ar ' 1 ' 16 ' 2.4. e* ~T 3 2 ' 2.4.6.8 X +64' 2.4.6.8.1O a; + ) 



Umvafi^./ 1 1 -JLL* 1 Jj. * 1-3.5.74, 45 1 . 3 . 5 . 7 . 9 5 , 5 5 1 . 8 . 5 . 7 . . ll^g , \ 

 9 ^ 3 2 ' 2 . 4 . 5* T"T' ITSTfTi* " t "512'2.4.e.8.10 1 *' "*"512'2.4.6.8.10.12 X T...; 



Ur>/i8/A/ 1 i-La_LAjJ~J--A- 1-3.5.7. 9 ^ , 33 1.3.5.7. 9 . lie , \ 

 0< P' Vl28 ' 2.4.6.8 1// """ase" 2.4.6.8.10'*' 1 "1024' 2.4.6.8.10.12 X T*J! 



+...} . . . :-. .... . '. y . v . I . . .-...-;; . (s). 



By inserting .r = 0'4 or S/a = 0'2254 in formula (8) this formula is reduced to 



r = 07746a(l-1318 + 0-1440 cos '2< + 0'0138 cos 4^ 



.) . . (9). 



The further calculations are based upon the following constants : V = 240 cm. /sec., 

 A = 0-066 cm 3 ., a = 0'1647 cm., p = 1, and T = 73 dyne/cm. The wave-lengths 

 corresponding to n = 2, 4, and 6, calculated by the above given means, are 



X 3 = 3'932 cm. ; X 4 = 1'227 cm. ; X = 0'646 cm. 



