■ ON STREAM-LINE MOTION OF A VISCOUS FLUID. 139 



formula used was only an approximation, though used within limits that 

 should give a very fair accuracy. Now it was not really necessary to use 

 the special formula for a channel at all, since if one of the cases be taken 

 in which exactly mathematical results can be obtained for an infinite fluid 

 — e.y., a cylinder — it is only necessary to form a border for a given value of 

 the stream function, and the test could be made and stream lines suitably 

 plotted within the artificial border. Glycerine is capable of being used at 

 very low velocities, and is absolutely steady in flow — indeed, owing to its 

 viscosity the lantern slide may be actually removed with the pipes dis- 

 connected, and after a lapse of even half an hour the stream lines remain 

 perfectly clear and distinct. It therefore seemed possible to make an 

 absolutely severe and final test, and the case of a cylinder and infinite 

 width of fluid were taken, the stream lines being plotted from the well- 

 known formula. This plotting was done by Mr. E. Brown, B.Sc, University 

 and 1851 Exhibition Scholar, who kindly undertook this particular work 

 for the author, besides rendering him valuable assistance with the 

 experiments for the present paper. 



On plotting the stream lines on a large scale, it was noticed that at a 

 distance from the cylinder corresponding to the distance at which colour 

 bands had previously been admitted, the stream lines were by no means 

 equally spaced, and moreover they were far from parallel to the direction 

 of flow of the fluid at infinity. To overcome this difliculty the stream 

 lines were extended to such a distance from the cylinder that they 

 became for all practical purposes parallel to the direction of flow at 

 infinity. The thin film slide was lengthened by a corresponding amount. 

 Further, it was noticed that at that distance the lines on the diagram, 

 which represent the theoretical stream lines, have a displacement from 

 the lines which correspond to the equal spacing of the stream lines only 

 actually attained at infinity. The colour bands were therefore admitted 

 to the film by holes which were so spaced as to correspond with the dis- 

 placement of the theoretical lines. 



The following three conditions were therefore introduced into the test 

 experiment, viz. — ■ 



1. Theoretical stream line as boundary. 



2. Longer film. 



3. Unequal spacing of stream lines. 



It was then found that the actual colour bands were in absolute 

 agreement with the theoretical lines. 



It must be remarked that in the previous verifications referred to the 

 author had been content to throw the lantern picture so as to most nearly 

 approximate with the theoretical diagram, which involved an obvious 

 displacement of the section of the cylinder itself, but in the present case 

 no such approximation was allowed in the border ; the obstacle in the 

 lantern itself was placed in absolutely exact position, so as to coincide with 

 the border on the plotted diagram. It need scarcely be said that this 

 result was not obtained without much laborious work, and it is highly 

 gratifying to know that the correctness of the result has been verified 

 mathematically in the accompanying paper by Professor Sir G. G. Stokes, 

 the conditions which it there states as necessary — viz., considerable viscosity 

 and very thin sheet — being both found necessary in order to obtain the 

 theoretically correct result. It should be noted that on the large scale in 

 which the comparisons were made by the lantern there is not the slightest 



