536 Mr DE MORGAN, ON SOME POINTS 



among the new variables as will eliminate the differentials, and leave only algebraic relations 

 between the new and old variables, to be combined with the assumed relations. In this case 

 arbitrary functions enter the final result. 



17. The primary solution of the biordinal partial equation contains five arbitrary 

 constants. Having given U(x,y,%,a,b,c,e,h)=0, we usually obtain a single biordinal 

 equation by eliminating a, b,... between the six equations U = 0, U x]z = 0, U y[z = 0, U m[t = 0, 

 77 _ 77 _ 0. But it is not true that every form of U gives one biordinal and one only : 



xy\z ' yy] % •* ■* 



there must be a certain irreducibility in the mode of entrance of the constants which does not 

 come under any obvious criterion. A case of exception may be easily given. Every primor- 

 dinal satisfies an infinite number of biordinals: and U(x,y, %, a, b, c, e, h) = 0, whenever it is 

 reducible to ( V, W, a, b, c, e, h) = 0, V and W being functions of x, y, %, and not of a, b, &c, 

 leads to a primordinal which is also independent of a, b, &c. 



Let us suppose, however, that U = leads to one biordinal equation. To obtain the 

 same equation by making a, b, &c. compensative, we must use d a>hceih U= 0, d ftKt)tH ^ tmu = 0, 

 d U , = 0. which are reducible to dc = Cda + C db, de = Eda + E, db, dh = Hda + H, db. 



a, b,c,e,h y\z ' ' i I 



First, suppose these equations integrable. Eliminate x,y,ss,h, between the five equations 



dc dc de de 



f/= °' da= C > db = C " da = E> db = E " 



... . . / . dc dc de de\ 



by which we obtain a, b, c, e, — , —-, — , -^7=0. 

 ' \ da db da db) 



Again, from any four of the five equations determine w, y, x, h, in terms of the rest, sub- 

 stitute in dh = Hda + Hdb, and then form the criterion {H) byce = {H} aU e , which is biordinal 

 with respect to c and e. If we can integrate jointly this last equation and (a, b, &c.) = 0, 

 then, in the integral of dh = Hda + Hdb and the five equations, we have the means of elimi- 

 nating a, b, c, e, h, and producing the general integral of the biordinal derived from U = 0. 

 In order, then, completely to characterize the general integral of a biordinal of three variables, 

 direct method requires that we should ascertain the integrals of two equations, one primordinal 

 and one biordinal, between four variables. 



Secondly, we attempt the question without requiring the equations of compensation to pre- 

 sent integrable forms. The primitives must be five equations between the eight variables 

 x, y, », a, b, c, e, h, and five new constants. These and their differentiations give ten equa- 

 tions, which elimination of constants reduces to five. By two of these and d x VZ U = 0, we 

 determine dx, dy, das, in terms of da, db, &c, and substitute in the rest, which gives three 

 equations between da, db, &c. ; and these must be jointly identical with the three equations of 

 compensation. Between the five primitives and U = 0, we eliminate a, b, c, e, h, and so get 

 another primary solution, namely, an equation between x, y, is, and the five new constants. 



18. The usual geometrical analogies give no assistance in determining the general form of 

 the primitive of a biordinal partial equation ; those of a less usual character, though jof no 

 greater success, will at least point out the conditions under which the solution is reduced to 



