Double Refraction in Liquids. 501 



of the force on one face of unit cube is proportional to the 

 expression 



2av (a + dV 



( a + d V 

 \ r ) 



2ad + a H ' y 

 in which 



v is the velocity of the moving cylinder ; 

 a is the radius of that cylinder ; 

 d is the distance between cylinders ; 

 r is the distance of the face considered from the axis of 

 the cylinder. 



In the case of the apparatus used, the concave wedges take 

 the place of the fixed cylinder. There is, midway between 

 the two cylinders, an infinitesimal layer of the liquid which 

 has zero velocity ; hence, the distance between the fixed and 

 the moving surface is d/2, and the stress on unit area tan- 

 gential to the cylinder is proportional to 



2av , , 7 ,~. 

 'a-t-a/2' 



?-(^-) 



ad + . 

 4 



or, since d/2 is small compared to a and r, approximately 

 2r/d. Assuming for liquids that which is found true in case 

 of solids, namely, that the accidental double refraction due to 

 a stress is proportional to the stress, we have, that the double 

 refraction varies directly as the velocity and inversely as the 

 distance between the surfaces. 



As a test of the correctness of this formula for the apparatus 

 used, observations were made with a solution of gelatine, and 

 the results obtained compared with those of other observers * 

 with two concentric cylinders. The results obtained agree 

 with those given by the other apparatus, so the experiment 

 confirms the correctness of the formula for the apparatus 

 used. 



Observations. 



With Water. — With water the most favourable conditions 

 are to be obtained. The sensitiveness attained with the 

 polariscope, with some training of the eye, is remarkable. A 

 rotation of the analyser which corresponds to a retardation 

 of one component with respect to the other of 0*000025 A 

 could readily be detected. Then water is typical of liquids 

 not oils, quite unlike colloids ; its chemical composition is 

 simple, its physical properties well known. 



* Umlauf, Ann. der Fhys. u. Chem, xlv. p. 304 (1892). 



