IN THE THEORY OF DIFFERENTIAL EQUATIONS. 533 



or augmentation, of the generality of the solution. The arbitrary function certainly appears 

 in every case in which the number of equations of compensation falls short of the number of 

 compensating variables : and the converse seems to be true. The difference between the two 

 numbers determines the number of variables in the arbitrary function ; that is, the number 

 of definite forms which may be arbitrarily combined : thus (p (xyz, x + y + z) is a function 

 of two variables, though each is a function of three. 



A system of polar curves in space will be a good illustration of the great differences 

 which exist between the laws of compensated systems. When Monge (Mem. Acad. Sci. 1784, 

 p. 502) undertook to give meaning to the equation f(x, y, z, y , z) = 0, y and z being func- 

 tions of x, he found, as he states, that such an equation as dz 2 = dx*+ dy 1 was considered as 

 absolutely absurd, and also Pdx + Qdy + Rdz = 0, except when integrable. This opinion 

 was founded, probably, upon the supposition that one equation must result from one primitive. 

 But it is easily seen, a priori, that a point moving so as to satisfy f(x, y, z, y, z') = 0, can- 

 not move at pleasure upon any possible surface, but may describe a curve upon any surface 

 whatsoever. This point, when at (I, m, ri) must begin to move along a generating line of 

 the cone f{l,m,n, (y — m):(x — I), (z — ri):(x — l)\ = : consequently, to suppose unre- 

 stricted motion upon some surface, is to suppose a surface every point of which has full con- 

 tact of the first order with some cone at the vertex. This can only be when the cone is a 

 plane, or when / has the form P + Qy'+ R% : and even then, as we otherwise know, only on 

 further conditions. Nevertheless, an infinity of curves may be described, the only condition 

 being that the curve shall pass through each of its points in the direction of some generating 

 line of the cone which has its vertex at that point. And such a curve may be drawn through 

 any point upon any surface, provided the tangent plane cut the cone in more than the vertex : 

 if not, there is still an imaginary curve, which satisfies the condition algebraically. This is 

 easily seen, in the usual way, from the equation itself. 



We want a term to express a surface composed, as it were, of curves, in the same manner 

 as we imagine a line composed of points, but coupled with the condition that we may not (or 

 at least do not) pass from one curve to another, though we may choose any one curve. The 

 word ruled is appropriated; but shaded is not. 



If ■% (x, y, z) = be obtained by eliminating a from (p (x, y, z, a) = 0, >J/- (x, y, z, a) m 0, 

 we may say that the surface ^ = is obtained by the mode of shading which is marked out 

 upon it by either (p = 0, or \|/- = 0. And thus we may say that the equation 



/(«> y, %, y, *') = 

 belongs to any given surface, provided one assignable mode of shading be adopted : with this 

 addition, that when the form is P+Qy'+Ez'=0 accompanied by the usual conditions of 

 integrability, there is one family of surfaces on which any mode of shading may be employed. 

 And conversely, any mode of shading may be chosen, provided one of the right family of sur- 

 faces be taken. 



15. Choose two equations between x, y, z, a, b, c, which reduce to y = (x, a, b, c), 

 z = -"p (x, a, b, c). Between these and y = $„ z ' - ¥,, eliminate a, b, c, producing 



f(x, y, z, y, z) = 0. 



