456 Cambiidge Philosophical Society. 



two points on a cun'e is the developable surface (or surfaces) drawn 

 through the two curves. The developable surface which touches 

 the given surface in a curve (and not the tangent plane) is the ana- 

 logue of the line which touches a curve in a point. A biordinal 

 equation being given, one surface satisfying it is selected by a curve 

 through which that surface is to pass, and a developable surface 

 passing through that curve which the surface is to touch. 



19, 20. The restrictions under which two arbitrary forms must 

 enter, in order that a biordinal partial equation may exist indepen- 

 dent of these functions, are wholly unknown. The case which is 

 fully analogous to a biordinal of two variables, is of the most limited 

 character. Ampere has noticed this : Mr. De Morgan was led to 

 it by an examination of the polar properties of <p(^x, y, z, a, b, c)=0. 

 This equation leads to a=A, 6=B, c=C, where A, B, C are func- 

 tions of X, y, z, p, q. The primordinal/(A, B, C) = is satisfied by 

 ^=0, subject to ({>(a, b, c)=0, and leads to a biordinal, independent 

 of/, of the form 



Q -H Rr + Ss + Tit -I- U(s9 - r<) = 0, 



in which Q, R, &c. are not wholly independent of each other. 



If the pole (a, b, c) move along a certain curve, the polar surface 

 must touch a certain surface in one of the lines of a certain shading. 

 That is, every aJc-curve has a shaded surface, which is its polar 

 reciprocal : and every line of shading of that surface has another 

 surface for its polar reciprocal, shaded by lines of which the original 

 flSc- curve is one. And every surface has a reciprocal surface such 

 that for each point on one there is a point on the other ; and the 

 point on one surface being taken, the polar surface of that point 

 touches the other surface in the other point. 



The singular solutions of the two biordinals derived from 

 <f>(x,y, z, a, b, c)=0 

 by means of x, y, z and of a, b, c, are connected by relations analo- 

 gous to those already seen in the case of two variables. In fact, 

 there is perfect coincidence and coextension between the properties 

 of the general equation y"=x(x,y,y') and a particular species of the 

 equation Q -j- Rr + Ss -|- T^ -(- 11(5^ — rt)=0. It is proposed to call 

 this species the polar biordinal. 



21. The general method of transforming partial equations, given 

 in the last paper, is the investigation of the class of surfaces con- 

 tained under a given equation by reference of them to their polar 

 reciprocals, any convenient modular equation (p(x, y, z, a, b, c) being 

 made the means of transformation. 



22. The following notation is proposed for eliminants. The com- 

 ponents being A^, A^, &c., B , &c., the eliminants are (A j, (AB ,^ 

 (ABC;,ryr), &c. ; the components being A, A', &c. B, &c,, the elimi- 

 nants are (A°), (AB° '), (ABC° ' "), &c. ITius 



(Ap)=A^ 



(ABpq)=Aj, (Bg^ — Bp (A, , 



(ABC^,,)= Ap (BC,,) + B;, (CA,,) + C^, (AB,,, 



(ABCD;,j,;=Ap(BCDj,,) -Bp(CDA,,,) +Cp(DABj„) -D;,(ABCj„), 



