570 



REPORT — 1891. 



li p be large enough, we can by taking a sufficient number of values of ot, 

 make the error of the order 1/;;^ &c. 



Roughly speaking, when the orifice is small compared -with the depth of the 

 liquid, the shape of the jet depends only on the orifice, being almost entirely 

 independent of tTie shape of the vessel. 



9. The Geometry of Confocal Conies. By Professor T. C. Lewis. 



1. If PP', QQ' be chords of contact of T in two confocal conies, then a conic 

 can be described with Q, Q' as foci which shall touch the conic PP' at P and P'. 



Fig. 1. 



2. By taking T on the outer of the two ellipses, it follows that the ellipse with 

 foci at Q, Q' which passes through T will have contact of the third order with the 

 confocal through T ; and the hyperbola with foci at Q, Q,' which passes through T 

 will have contact of the third order with the hyperbola whose foci are S and H 

 which passes through T. 



3. In fig. 3 PQ, PQ', P'Q, P'Q' are all tangents to one ellipse confocal with 

 that through Q, Q'. Let this be q^ q„ q^ q^ (fig. 3). The four tangents to an ellipse 

 intersect in three pairs of points. If each of one pair is on a confocal conic, so also 

 is each of the others. Q, Q' lie on one confocal ; T, T' lie on another confocal. 



4. As in 1, a conic with P, P' as foci can be drawn touching the confocal TT' 

 at T and T' .-. PT-TP' = PT'-TT'. 



.'. a circle can be inscribed in the quadrilateral TPT'P'. 



Fig. 3. 



Fig. 4. 



5. As a particular case, when P and Q coincide the two tangents from T meet 

 the tangent at L in Q, Q' (fig. 4) which are on a confocal, and the circle inscribed 

 in TQQ' touches the ellipse at L. 



6. In fig. 3, QP + PQ' = QP' + P'Q'. Add to each Qy, + arc q,q^ + y.Q' ; and let 

 PP' be consecutive points ; Q being the intersection of consecutive tangents lies on 

 the inner ellipse. Hence the string which must be placed round the inner ellipse 

 to stretch to P is equal to the one which must be placed round it to stretch to P'. 



