Prof. Magnus's Hydraulic Researches. 179 



which is depicted in fig. \7 a, as seen by an observer who stands 

 in the prolongation of xy. 



From the aperture xyf yn y' the water falls down in four arms 

 which cross themselves, y^ep ye, y^e^,, &c., of which each has a 

 strong edge. By the encounter of these four edges, every two 

 form a surface, ?'pq. Since, howevei", the encounter is central, 

 each surface bisects the angle formed by the edges by which 

 they were made. The four surfaces formed in this manner have, 

 hence, the position, fig. 17, marked by the dotted lines pp. They 

 are at right angles to each other, and bisect the angles of the 

 ci'oss yy,y„y'. The edges ye stretch between them down to g. 

 In the horizontal line going through this point, that is at p, the 

 surfaces epq have their greatest breadth. Below this part they 

 again assume thick edges, by whose encounter new surfaces, zv, 

 are produced. Since the edges 2^1 find p^q, &c. also encounter 

 centrally, the new surfaces zv, &c. bisect again the angle which 

 the edges j^^r, &c. form with each other, and are hence again 

 at right angles to each other, and have the same situation as the 

 arms ye, y^e^, y^^e^, of the cross. 



I have seldom succeeded in obtaining more than twQ such 

 systems of surfaces under each other ; perhaps because the efflux 

 orifice was not sufficientljr accm-ately made ; for the smallest 

 diflFerence in the dimensions of the individual arms ocy or xy^ 

 deranges the symmetry of the surfaces rpq and their edges. The 

 formation of a third and fourth system of sui'faces is likewise 

 rendered more difficult, fi'om reasons similar to those to be 

 afterwards mentioned in § 83. 



62. If any of the before-mentioned disturbances occur in the 

 efflux, the edges no longer encounter centrally, and the jet 

 assumes a spiral form. There are, however, few cases in which 

 such a spiral form is regular, for it is seldom that the disturb- 

 ances act symmetrically on all four arms of the cross yyiy^y'' 

 Fig. 17 c represents this form as it was obtained by placing 

 pieces of metal plate E exactly of the same size, and in the same 

 position on the four arms ye, yf^, y„<?;,, y'e! of the cruciform aper- 

 ture, fig. \7 b. If these pieces were not all exactly alike, or if 

 they were not all placed symmetrically, the jet had not a regular 

 spiral shape. The metal pieces E must not be too thick, for 

 then the individual windings yp, yiP^, y'p', &c. would separate. 



63. IIow remarkable the form of the jet issuing from the 

 cruciform aperture may be, and yet how similar to the phseno- 

 mena hitherto observed, is seen when the efflux from one of the 

 four arms of the cross is stopped, so that the efflux orifice has 

 the form y, y„ y^, fig. 18. The jet assumes, then, the form 

 represented in fig. 18 « and fig. 18 6. Fig. 18 « represents it as 

 Been by an observer standing at right angles to yxy^, and fig. 18 b 

 as seen by one who stands in the prolongation of yxy,. 



