ON STREAM-LINE MOTION OF A VISCOUS FLUID. 141 



The converse effect can be produced by making a part of the thin 

 sheet rather deeper than the remaining portion. This of course will 

 correspond to the flow through an obstacle of similar shape, which opposes 

 less resistance than the surrounding space. The effects are obviously the 

 same as would be produced in the first case by a dia-magnetic body, and 

 in the second case by a para-magnetic body, and by varying the relative 

 thickness of the different portions of the sheet it is clear that the effects 

 which would be given by the body of any known resistance (i.e., of any 

 value of n) can be produced. 



The author at first attempted to produce these results by very thin 

 sheets of glass. It was seen that where the stream meets the edge an 

 effect corresponding to refraction is produced, but that while in the case 

 of the plate touching at both edges, although the velocity must obviously 

 be greater over the portion where the plate is partly obstructing the 

 channel, that is, makes the channel rather shallower, the width of the 

 colour bands remains the same. When, however, the obstacle does not 

 touch the edges, the effect is to produce very much wider bands over the 

 obstacle itself and narrower bands on either side, these bands giving an 

 indication of the great difference of velocity which results from the 

 greatly increased resistance over the surface of the obstruction. 



Instead of considering merely the actual width of the bands, it is of 

 course possible, and generally more convenient, to consider the number 

 of bands in a given space. This method is applied to a circular hole in a 

 plate of different cross sections to the film itself, which is placed across 

 the entire width of the channel, and the number of bands or stream lines 

 in a given space in the hole or well is obviously greater than in the 

 surrounding portion — i.e., the bands are close together, and the velocity 

 correspondingly greater. This result evidently represents the effect of 

 placing in a uniform magnetic field a circular cylinder of soft iron — i.e., 

 a para-magnetic body. This sufiiciently indicates the method, but other 

 examples may be given, the first two representing para-magnetic and dia- 

 magnetic cylinders, which are cases which can be dealt with by means of 

 mathematics ; also two other cases of cylinders of rectangular section, 

 representing respectively the result with para- and dia-magnetic bodies, 

 which are cases it has been hitherto impossible to deal with by mathe- 

 matical methods. 



(6.) The Effect of Using a Wedge-shape Section. 



The author attempted to solve the problem of obtaining the flow 

 round a solid of revolution by using a wedge-shaped section, the obstruc- 

 tion being also represented by a wedge representing a segment of the 

 body, the thinnest part of the wedge corresponding to the axis of revo- 

 lution. 



Professor Stokes has been good enough to look into this matter, and 

 has found that the partial differential equation which the stream-line 

 function must satisfy in the case of a slender wedge of viscous fluid is 



dx' dy"^ y dy 



X being measured parallel and y perpendicular to the edge ; whereas, for 

 a perfect fluid flowing axially over a solid of revolution, generated by the 



