507 



(c) The Shape Co-efficients of the BuPDIe :- If R be the radius vector from a point on the 

 axis of a buPble to its surface, then a bubble of any given shape (assuming axial sytmetry) may be 

 represented by an infinite set of "shape co-e-fflcients" as follows:- 



R = a ♦ bjP^ (cos 9 ) ♦ b^P (cos 6) + etc. 



where b b etc. are here called the "shape co-efficients" 



6 is the angle between the radius vector and the axis of symnetry; 9 = o is tal<en downward. 

 Pn is the Legendre Polynomial of the nth order. 



To a first order in small quantities, b. will be zero if the origin of the co-ordinates is 

 tal<en at tho centre of gravity. since it was desired to compare the observed shape co-efficients with 

 those calculated theoretically in which the origin is usually chosen so as to make b always zero, the 

 centre of gravity was chosen as the origin for measurements. 



The method of measuring the co-efficients a, b , b vtc. for a given bubble outline is considered 

 in detail in an Appendix. Briefly, the method assumes that no co-efficients higher than b are necessary 

 to express the shape observed. Radii vectores are measured for seven equally spaced values of 5, .and the 

 resulting simultaneous equations are solved for the seven co-efficients. It is sufficient justification 

 of the method, here, to say that when the full outline given by the seven co-efficients thus calculated 

 is drawn cut it agrees with the observed outline with an error less than the uncertainty in delineating 

 the. observed outline cf the bubble, even for extreme cases so far found. 



(d) Accuracy of ^'easure^ne^t :- Length measurements can be made on the photographs within about 

 2 or 3J, but absolute values cf the position tj the bubble are probably only accurate to within about 

 i inch. Values cf the velocity ;f displacement of the bubble are therefore subject to rather large 

 inaccuracies, though this is somewhat reduced by drawing a smooth curve tnrcugh a number of observations. 



Experimental Results . 



The experimental results are plotted in figures i to 5. The timing of all observations near 

 a minimum was measured in milliseconds before or after the minimum rather than in milliseconds after 

 detonation since there was considerable variation in the value of the first pe'riod. The period varied 

 between 7it and 80 milliseconds with an average of 77 milliseconds, and this value has been used in 

 plotting the data in Figure i. 



The data for the second oscillati 



ife scanty and have been put in a table which follows. 



Description of the Photof^rap ks . 



In Figures 7. 8 and 9 a number of typical photographs of the bubble are given in each of which 

 the bubble appears silhouetted against a white reflector. The explosive, being of a type which gives 

 rise to no free carbon in the explosion products, resulted in a bubble which was fairly transparent so 

 that in most photograph? a splash of light appeared at the centre of the bubble where the gas-water 



