264 



Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23, 



same set, there are thirty-two groups of four conies formed by 

 taking two out of any one set and two out of any other, such 

 that each group is touched by two conies having double contact 

 with the given one </>, besides the two they have been constructed 

 to touch in common, and as the four sets can be taken in pairs in 

 six ways, there are thus one hundred and ninety-two such groups 

 in all. 



Some other theorems are obtained of a more complicated 

 character. 



The method of proof is purely geometrical and the first pro- 

 position, though not proved as shortly as it might have been, is 

 made to depend mainly on a property of Bicircular Quartics 

 given by Mr C. M. Jessop in the Quarterly Journal of Mathematics, 

 Vol. XXIII., which is equally true for Sphero-Quartics, and which 

 for the case of two circles may be enunciated a little differently 

 as follows : — If X, Y are any two circles of the same family 

 touching two given circles A and B, in a plane or on a sphere, 

 and P, Q are any two circles of the other family touching A and 

 B, then a circle can be drawn through one of each pair of the 

 intersections of X with P, X with Q, Y with P, and Y with Q, and 

 of course another circle through the remaining four intersections 

 of the same pairs. 



(4) On Liquid Jets under Gravity. By H. J. Sharpe, M.A., 

 St John's College. 



1. The motion (Fig. 1), which is in two dimensions, is sup- 

 posed to be symmetrical with regard to x'Ox which is the axis 

 of the vessel and jet. BEF is the semi-outline of the vessel, 

 FJ of the jet. AF is the semi-orifice, which is small compared 

 with the dimensions of the vessel and the depth OA of the 

 liquid. Gravity acts parallel to x'Ox. OE is the surface of the 



liquid, which is maintained steady. AF is supposed to be so 

 small that it may be considered either as the arc of a circle 



