568 BEPOET— 1891. 



6. On the Transformations used in connection with the duality of Differen- 

 tial Equations. By E. B. Elliott, F.E.S. 



The Mouge-Chasles-De Morgan reciprocal transformation of partial differential 

 expressions 



x' = p,i/' = g,z' = px + g7/-z, 



x = p',y = q',z=p'3:' + q'y'-z\ • 



is readily carried beyond the second order of derivatives by noticing that what is 

 required is to express the derivatives of j), q with regard to p', q' in terms of those 

 Q'i p , q' with regard to ;>, q. Now a theory of the reversion of partial derivatives 

 of two variables with regard to two others by interchange of the dependent and 

 independent pairs has been developed. 



The analogous but simpler reciprocal transformation of ordinary differential 

 expressions 



x'=p,y'=px-y 



or 



x=p',y=p'x'-y' 



amounts only to the interchange of dependent and independent variables in deriva- 

 tives oi p with regard to^j'; and a quite complete theory of such a reversion is at 

 our disposal. One consequence is that any reciprocant gives us on replacing 



-/-, --^.., . . , by ~, -=-4 .... a self-reciprocal expression, i.e.. the criterion of 

 dp'^ dp^ d.i" dx^ 



a family of curves whose polar reciprocals with regard to the parabola x" = 2y con- 

 stitute the same family. 



7. Note on a Method of Research for Invariants, 

 By E. B. Elliott, F.B.8. 



This note was of the nature of an inquiry as to whether adequate use had been 

 made of methods of direct determination of invariants of a binary form in terms of 

 its co-eilicients when deprived of its second term. The invariants of 



w(n-l) „_j w(w-l)(«-2) ,_3 

 ax" + \i) ex + Y2S + . . . 



are as shown by Cayley those functions of a,c, d, . . . whose degree i and weight w 

 satisfy in ■= 2w, and which are annihilated by the differential operator 





8. On Liquid, Jets under Gravity. By Rev. H. J. Shaepe, M.A. 



The motion, which is in two dimensions, is supposed to be symmetrical with 

 regard to x'dx, 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 of the liquid. Gravity acts parallel to 

 .r'O.r. 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 with 

 centre in the surface of the liquid, or as a small straight line perpendicular to 

 O.r. For simplicity we shall take OA the radius of the circle (or the depth of the 

 liquid) as unity. If </ be the acceleration of gravity referred to this unit, it will 

 be convenient to put a- for 2g. We shall take O as the origin of Cartesian and 

 polar coordinates x, y, r, 6 and we shall put x' for (.r — 1). Let x ^"d t//- be the 

 stream functions on the right and left respectively of AF and let u, v be the 

 velocities parallel to Ox, Oy. Further let AF = irip where ^ is a large number. 



