IN THE THEORY OF DIFFERENTIAL EQUATIONS. 539 



before I had examined his paper. Let the primitive equation be qy(x,y, x, a, b, c) ■ 0. It is 

 not necessary formally to enunciate the definition of (a, b, c) as the pole of the t ry«-surface 

 (p m 0, &c, nor to state the theorems with which the geometrical reader is familiar in the usual 

 case. By = 0, (p xlz =0, <p nz = 0, we produce a - A (x, y, x, p, q), b = B{), c=C( ). 

 These equations must identically satisfy <p a dA + (p b dB + (p c dC = 0: for <p = gives 

 d xyz <p + d a b tC <p => 0, and a = A, &c. satisfy <p = 0, d x y z d) = 0, identically. If then one of 

 the three, say A, be made constant, we have dB:dC = - (/><.:<£,, and is primordinal. But 

 dA=0 or A x[z>pq dx + A ylzpq dy = 0, substituted in dB:dC, gives 



di? J ti B ti -A ti B tl= <p c 

 dc" C.iA wX -C, x A, x ft* 



the two last functions becoming identical when A, B, C, are written for a, b, c. It follows 

 that A ki B fl - A 9X B,p B x C,, - B yi C x , C,,^, - C 9X A tV have a common biordinal factor, all 

 factors not common being primordinal. Hence \|/ (A, B, C) = is a primordinal equation 

 which has but one biordinal consequence independent of \|^. From it we deduce 



which is satisfied, independently of \^, by the biordinal factor just deduced. Now \L = has 

 for a primary solution (a?, ^, a?, a, 6, c) = 0, subject to \js (a, 6, c) = : but we are not to 

 conclude that the primitive of the biordinal contains an arbitrary function of two variables ; 

 for C x A yi -C y ^A x] , A x[ B y{ -A y] B xj , B x[ C y{ - B y[ C x[ severally vanish under C = (3A, 

 A = yB, B = aC, (3, y, a, being any forms whatever. There are then two primordinal equa- 

 tions of the form y\r(A, B, C) = 0, reducible to A = yB, B = aC. And <p(co, y,z, a, \a, pa) = 

 is a solution of B = \A and of C = /nA, as well as of their common biordinal. If a be made 

 self-compensating, by derivation from <p a + d) b \'a + (p^'a = 0, the primordinals remain true, 

 the biordinal follows as before, and substitution of the value of a in <p = gives the complete 

 primitive of the biordinal. 



This biordinal itself is of a very limited form. From B = \A we obtain 



4,B, x -A 9l B, = 0, or 



(A, + A z p + A p r + A q s)(B y + B z q + B p s + B q t) - (A y + A z q + A p s + A q t){B x + B z p + B p r + B q s) - 0, 



or Q + Rr + Ss + Tt + U(s* - rt) = 0, 



where Q, R, &c. are functions of as, y, %, p, q. But it is only a certain species of this 

 general form which is fully analogous to the biordinal of two variables : and this species 

 may be called polar. 



If the pole (a, b, c) be placed upon the curve b = Xa, c = /u.a, the polar ,ry#-surface is made 

 subject to B = XA, C = /uA. If the pole move along that curve, the polar surface always 

 touches the surface obtained by eliminating a from <p = 0, <p a + (p b X'a + (p^'a = 0, and touches 

 in the whole curve which has these two equations. That is, every a&c-curve has a shaded 

 surface which is its polar reciprocal ; and if the pole (#, y, *) be made to move along any 



