40 DISPLACEMENT INTERFEROMETRY. 



Tubes of the type I are thus unsatisfactory. In case of tubes of the types 

 II or III the thermal displacement would be but about 3 to 4 per cent of the 

 elastic displacement in case of water; but in case of alcohol 30 and 24 per cent; 

 of ether 47 and 38 per cent. In tubes of the type IV the advantages of thicker 

 walls and small external diameter are further in evidence : alcohol and ether 

 show ratios of 38 and 59 per cent. The problem of selecting the best tube 

 admits of general solution, to which these data contribute. 



If we combine equations (4) and (8) and put 



A=Al/l B = Ap/$k C = p'c'/pc x(rz z ri z }/ri 



the result is 



A-B/x 



(9) 



i+Cx 



Here x is the ratio of solid and liquid sections and A the original total elonga- 

 tion. We inquire what value of x will make A0' a maximum provided A, B, C 

 are constant. If the original thermal and elastic elongations are to be equal 

 A = -2.B. Differentiating (9) and reducing : 



(10) - = C( 



oc 



and since x must be positive the radical is positive. Now if A = 2B, for example 



(") - 



or the ratio of diameters zr\ to 2^2 would in all cases have to exceed 0.65. 

 If A^B, the case of water remains nearly the same, but for ether and alcohol 

 the diameter ratio approaches 0.9. Massive tubes implying complications 

 owing to heat conduction are to be avoided so far as possible. 



