538 



Mr DE MORGAN, ON SOME POINTS 



of the second order, the relations between p and q which exist on the next line of shading, by 

 which, with like errors, we determine the tangent developable of that second line ; on which 

 we may repeat the process. It is thus shown that a surface is determined when we know one 

 curve, the tangent developable belonging to that curve, and a biordinal partial equation 

 which belongs to the whole surface. This is an a priori proof that two, and no more, rela- 

 tions of form are requisite for the determination of each surface in f = 0. 



Let <p (x, y, ss) = be the complete primitive, and y = ax an assigned projection on my : 

 let x = fix then determine the curve, and q = Fp the developable. Then (p (x, ax, fix) = 0, 

 - (p v :(p z = F(- (p x :<p z ), independently of x ; from which the forms of /3 and F are to be 

 determined. The conditions of the question thus make it apparent that algebraical expression 

 of a geometrical construction which certainly determines one surface, gives conditions enough 

 to determine two unknown forms of function : but this does not show that any given mode of 

 entrance of arbitrary functions is consistent with an equation between x, y, *, p, q, r, s, and t, 

 independently of the forms of those functions. 



19. The restrictions under which two arbitrary functions must enter, so as to neces- 

 sitate a biordinal equation which is free of those functions, are wholly unknown. The simple 

 relation % = x<py + \j/ {x + y) does not succeed : but* if 



3x m v 4 x — 2v*y + 4 fv"(p'vdv + 3\{/ (vx + v~ x y + <pv), y = v 2 (x + <p'v), 



we may deduce r — pt = q ^/rt. The mere appearance of an arbitrary form entering the 

 subject of another, and the possibility of a continuance of this mode of entrance, as in 



^[A + ( p{B + f(C+(l)(D+ &c.))}], 



joined to the failure of every attempt to obtain the general form, may lead us to suspect that 

 the general integral and the biordinal equation have what might be called a functio-functional 



relation, or a functional relation of the second order. I mean such a relation as / <px dx bears 

 to <px : since the first depends wholly on the second, we may write 



/ (f>x dx = {F . (p) x, 



but not F . ((px) ; F brings out a form depending on the form of <p, just as (p in <px brings 

 out a value depending on the value of x. 



20. It can be shown that the case which is fully analogous to the most general biordinal 

 equation of two variables is not the general biordinal equation of three variables, but a very 

 limited instance of it. This result belongs to Ampere : but the polar relation led me to it 



* This instance is taken from a paper by Ampere on partial 

 differential equations, in the Journal de I'Ecole Poll/technique, 

 vol. x. pp. 549-611, and vol. xii. pp. 1-188. Few have had 

 the courage to read this voluminous paper, the notation of 

 which appears, though it really is not, unusually intricate. It 

 would be valuable to any investigator who is concerned with a 

 case in which one variable is in two ways a function of two 



others, one of which is in both, as in z = <j> (x, y) =ifr(x, t). 



Ampere denoted the two pairs of differential coefficients thence 



resulting by 



de dz , dz , dz 



, , , . and -r—r- and -r— 



dx(y dy(x dx(t dt(x 



of which it is more easy to see the cumbrous effect than the 



means of avoiding it which would secure general assent. 



