IN THE THEORY OF DIFFERENTIAL EQUATIONS. 543 



Then, since (CCDEF Mr3t) = 0, &c, we have 

 (ACDEF^.a + iBCDEF^.fi-o, or (CDEFA^.a + (BCDEF^fi = 0. 



And thus it will be found that the six quantities a, /3, <y, &c. are in the proportion of the 

 sixth eliminants of the fifth order (BCDEF mrsl) , (CDEFA^, (DEFAB^, &c, with the 

 alternate signs which they take in the formation of one of the sixth order. 



In writing, and even in printing, the parentheses of the symbol (ABC , or (ABC 



i in 



may be omitted. The two sets of symbols having the same number in each is a guide 

 sufficient to prevent mistake. 



23. I now return to Q + Rr + Ss + Tt + U (s* - rt) = 0, which includes the polar form 

 of the biordinal equation, and which must follow, independently of "ar, from 



B (x, y, z, p,q)=w {A (w, y, *, p,q)). 



Some cases have no such primordinal solution, some have one only, and some have two : these 

 last are polar, that is, have a solution of the form <p (x, y, z, a, b, c) = 0, which may be called 

 the polar primitive. The biordinal itself, derived from B = tstA, is 



A H —A II — n 



■™x\z,p,q "y\z,p,q ■ a y\z,p,q ■"x\z,p,q — v > 



or {AB m + {AB m r + \(AB qy) + (AB xp) \ s + (AB^ t + (AB qp) («« - rt) = 0, 



where (AB xy) means A x B y - A y B x , &c. But A x and A y signify A t \ x and A yv or A x + A z p 

 and A y + A z q. 



Let B = srA and D = kC be two primordinal solutions of one and the same biordinal. 

 We have then 



AB m -M.CD„, AB py = M.CD py , AB iq = M.CD xq! AB qp = M.CD qv , 



AB v + JB„i>MiCD v +CDJ, 



M being M (x, y, z, p, q). Consequently, 



AB xq AB m + AB^ AB qp = M °- (CD^ CD^ + CD„ CD qp ) 

 or* AB^ABv-M'CD^CD,,. 



From this, and the fifth equation preceding, we see that either AB = M . CD X and 

 AB m = M. CD W or else AB^ = M . CD qy and AB qy = M. CD^. In the first case, all the six 

 AB eliminants are proportional to the six CD eliminants ; whence (ABCD xypq) becomes 2M 

 multiplied by AB^ AB^ - AB^ AB yq + AB iq AB yp and vanishes. Now (ABCD xypq) = is the 

 criterion which shows that f (A, B, C, D) can be identically satisfied by some form of /, x 

 entering only as a function of x and y. But we shall presently discuss this result in a more 

 satisfactory manner. 



If we now look only at B = "&A, as a solution of Q + Rr + &c. = 0, we have AB^ = MQ, 

 AB m = MR, &c, whence we get, as before, AB^ + AB qy = MS, AB^ AB W = M 2 (RT + QU). 



* The two following equations are most essential in what follows : 



ABCD„Kd= (AB& CDcd+ ABdCDab) - {ABm CDbj + AB U CD ar ) + (ABad CD ic + AB bc CD„j) 

 AB„k AB cd = AB M AB U + AB ai AB cb . 



Vol. IX. Part IV. 70 



