IN THE THEORY OF DIFFERENTIAL EQUATIONS. 545 



The particular cases A = const., B = const., are solutions. If in B = wA, we assume t&A 

 to be a function which approximates towards a constant, then A x{ B y — A yX B xX will have 

 a factor independent of r, s, t, which diminishes without limit. But if a division by this 

 factor be made, the quotient, Q + Br + &c. is always = 0, absolutely, when B = srA. This 

 then is true at the limit, or when B = const. But, though A = const, and B = const., are 

 solutions, it does not follow that they can coexist as solutions. Thus in the preceding 

 example, py = a, qx = b, are solutions, but cannot coexist; for ay~ l dx + bx~ 1 dy m dz is not 

 integrable. Coexistence requires that s, as obtained from p in A = a, B = b, should agree 

 with s as obtained from q. Now, A x still meaning A x + pA z , &c, we find from 



A x + A p r + A<p = and B x + B p r + BqS = 0, that ^# rp + AB qp . s = ; 



we also find ^.Bj, + AB pq ,s=0. Hence AB xp = AB qy is the condition necessary to the 

 coexistence of A = a and B = b. Hence k ■» 1, the two values of & are equal, and 

 (S 2 = 4(^7 T + Q£7) is a necessary equation of condition between the coefficients of the 

 biordinal. 



We cannot have two coexisting primordinals P {A, B) = 0, Q (C, D) = 0, unless 

 A = const., B = const., G = const., D = const, can coexist. That the first and second pairs may 

 coexist is obvious from the equations P = 0, Q = 0. Again, the coexistence of these last 

 equations gives 



PQ rp = PQ qy or P x Q p - P p Q x = P,Q y - P y Q q . 



Writing for P x its value P A A X + P B B X , &c. we find 



^Gc (^ - AC J + P B Q C (AC,, - JBC W ) + P,Q D ( J D ip - ADJ + P B Q D {BD ip - BDJ = 0. 



This cannot be always true, independently of the forms of P and Q, Unless 

 AC^^AC^, &c. 



It is also clear that four distinct solutions A = const. &c. cannot coexist, unless either of 

 the four, A, B, C, D be identically a function of the other three. For, otherwise, elimination 

 of p, q, x, would give a relation between the independent variables x and y. 



When Q + Br + &c. is polar, that is, derivable from (p (x, y, x, a, b, c) = 0, there are 

 three coexistent solutions A = a, B = b, C = c. Consequently S* = 4 (BT + QU) is a 

 necessary condition. And if three such solutions can be found, from the equations marked 

 (v), it follows that B = wA, C = kB satisfy Q + Br + &c. = 0. Substitution of A and B in 

 (w) will give four equations, from which common elimination will restore the four relations 

 seen in the five equations AB y = MB, &c. It may be easily verified that A = a, B = b, C = c, 

 when satisfying (v), give a polar primitive by elimination of p and q. The relation between 

 x, y, x, a,b, c thence obtained gives A s + A z x x + A p x + A q q x = 0, and similarly for B and 

 C : in order then that *, = p, we must have A x + A z p + A p p x + A q q x = 0, or, making the 

 abbreviation hitherto used, A x + A p x + A q q x = 0. Eliminating p x and q x from the three 

 equations thus obtained, we have ABC xpq = : and similarly we obtain ABC ypq = 0. These 

 equations, resolved into A x . BC pq + &c. = 0, A y . BC pq + &c. = 0, give AB xy , BC xy , CA xy , 

 proportional to AB pq , BC pq , CA pq , which is known to be true, the proportion being that of 

 Q to - U. 



70—2 



