544 Mr DE MORGAN, ON SOME POINTS 



Let AB m k . AB ; then kS* = (l + ky(RT + QU). The five equations of connexion may 

 now be replaced by 



AB py = MR, AB^MT, AB V = MU, AB jp = -^-MS,AB gv = 7 ±-.MS, 



from which AAB xpq = 0, AAB m = 0, give 



UA ^ T ^-J^- k SA q = o, UA^RA^-^SA^O, 



in which B may be written for A, and k (which must be the same in both equations) is either 

 solution of kS 2 = (l + k)' 2 (RT + QU). Hence the following theorem; — To determine the 

 primordinals, if any, of Q + Rr + Ss + Tt + U (s 2 - rt) = 0, in which an arbitrary function 

 explicitly occurs, as in B {x, y, z, p, q) = tst A (on, y, z, p, q), find separately, by common 

 methods, the complete solutions of 



[dv dv\ m dv k dv 



U _ + p — \ + T S— = 0, 



\dx das] dp 1 + k dq 



(dv dv\ dv l dv 



f7 b-+9T" +R -3 7 S T = °- 



\dy dzl dq 1 + k dp 



(») 



a, y, x, p, q being five independent variables, and k being one of the roots of 



feS^-O +k)*{RT+ QU). 



If A and B be any functions of a?, y, ss, p, q, each of which satisfies both equations, then 

 B = "ST A is one primordinal solution required. If no such common solution, or one only, 

 should exist, there can be no primordinal of the form required. 



For example, let pq + (px + qy) s + my (s 2 — rt) = 0. Here k is either pw : qy, or 

 qy : pw. We have, for the first, 



dv dv dv 



y^- + yP^--P^- = > v = d>{y,p,z-px,px + qy), 



dw ass dq 



dv dv dv , , . 



m — + wq — - q — = 0, v = -d, (w, q, z - qy, px + qy), 



dy dz dp , 



of which v = px + qy is the only fundamental common solution. The second value of k 



gives 



dv dv dv 



x — + xp — - q — = 0, v = <p(y, p, qx, z - px), 



dx dz dq 



dv dv dv 



y dy + qy dz-~ P dp~ = ' v = ^(x,q,py,z- qy), 



of which v = qx, v = py are fundamental common solutions, and the only ones. Hence the 

 given equation has a primordinal solution of the required form, qx — ■& {py), and no other. 



