Similarly, if $, is the angle contained between the normal at P, and the plane P, PN, Hence as $i = 02, equating (187) and (188), we have which is the condition of integrability; and therefore we infer that if (179) does not satisfy this condition, it does not express the property of a surface, and its integral cannot be of the form u = F(x, y, z) = c. If the point p on the surface is taken for the origin, and the normal PN is taken for the z-axis, and the lines PP, and PP, are taken for the axes of x and of y respectively, then $1 = n2 = do, ni = $i = $2 = %2 = 0; so that. sin (191) whereby we have a geometrical interpretation of the ordinary criteria that the equation p dx + Q dy = 0 is an exact differential. I may also observe that if the axes of coordinates, to which the surface supposed to be represented by the differential equation Pdx +qdy +Rdz = () is referred, are transformed into another rectangular system, in which the several letters are accented, it may be shewn that (190) becomes so that this condition is invariant; a result which may generally be inferred from the circumstance that it expresses a geometrical property of surfaces which is true independently of any particular system of reference. 401.] We now return to the analytical investigation. It appears that an equation of the form pdx -+ Qdy = 0 can always be rendered an exact differential by means of a multiplier, and that its integral involves an arbitrary functional symbol; and it also appears that pdx+Qdy + Rdz = 0) is not always capable of being made an exact differential by means of a multiplier, and can be made so only when the condition (166) is satisfied. If however (166) is not satisfied, but pdx + Qdy + Rdz can be separated into two parts, which are respectively exact differentials multiplied by factors, so that it is of the form H DU +M, DU, = 0, (193) it is evidently satisfied by v = 0, , = C1; u and y, being so related, that both are simultaneously constant; and therefore | U1 = (U), (194) the form of being at present undetermined: but as from (194) DU, = $'(U)DU ; substituting this in (193) we have A+B+ (U) = 0; which equations are sufficient for determining the form of $; and the result becomes V = C, " = " (C); each of which is the equation to a surface; and the two when taken simultaneously, as it is necessary in this case, express the curve of intersection of the two surfaces : the differential equation therefore expresses a property of a curve and not of a surface. Or again if we cannot by inspection separate the differential expression into two parts of the form (193); yet by the following process we can shew that it expresses a property of a curve and not of a surface; that is, if (x, y, z) is a point on a surface, it is possible to draw through the point and on the surface an infinite number of lines, the consecutive points of which shall satisfy the differential equation, although the equation to the surface does not. For suppose the equation to the surface to be F, (x, y, z) = G; whence we have u dx + v dy + w dz = 0; then multiplying this last by v, and adding it to the given differential equation, (P + v U)dx +(Q+vv)dy + (+vw)dz = 0; (195) now suppose v to be so determined that this shall satisfy the condition (166): let the integral of (195) be F, (x, y, z) = Cz; then F, and F, taken together satisfy the differential equation; and therefore all the curves in which these two surfaces intersect satisfy the equation : now F, will manifestly contain an arbitrary function, and therefore there will be an infinite number of lines of intersection; although therefore no one surface satisfies the conditions of the given equation, yet through any point on the surface F, may an infinite number of lines be drawn along which we may pass without violating the conditions, but we are unable to pass from one line to another across the others. Another way of considering the matter is this ; assume y = $(x), and substitute for y in the given differential equation; have {P+Q¢'(x)} dx + R dz = 0, (196) $(x) having been substituted for y in P, Q and R. Suppose the integral of (196) to be be F(x, z, C) = 0, where c is an arbitrary constant: then the intersection of the cylinders whose equations are y = $(x), and F(x,z,c)=0, satisfies the requirements of the given differential equation. Let y = 0(x), so that dy = $'(x) dx ; then the equation becomes z dx + x $'(x) dx + (x) dz = 0, {+ x $'(x)} dx +0 (x) dz = 0; the integral of which and the equation y = $(x) together satisfy the differential equation. Thus if y = x +c, then (x+c)dz +z dx + x dx = 0, (x+c)2+ = @ Now this equation is that to a hyperbolic cylinder perpendicular to the plane of (x, z): and y = x+c expresses a plane perpendicular to the plane of (x, y): and each of these involves an arbitrary constant; consequently the series of lines of intersections of these two surfaces satisfy the given differential equation. I have said nothing as to the means of determining the integrating factor of a differential expression of more than three variables, because I am unwilling to enlarge the volume by investigations which are not necessary, aids in our subsequent applications of pure mathematics to physics. SECTION 7.—On singular solutions of differential equations of the first order. 402.] Thus far we have investigated general and particular integrals of differential equations of the first order ; but in some cases there are functions of x and y which satisfy the given equation, and yet cannot be deduced from the general integral by any particular value of the arbitrary constant: such functions are called singular solutions, as we have already noticed in Art. 368, and we now proceed to investigate their properties and modes of discovery. As the inquiry is one of the most difficult in this branch of our subject, the best course is to recur to first principles of definite integration, and thus to state the question and its conditions in the most elementary form. Let us assume the differential equation, whose singular solution is required, to be exact, and to be of the form P dx + dy = Du = 0, (197) PRICE, VOL. II. 4 D where P and Q are functions of x and y; and let us replace by f (x,y); so that dy = f(x,y) dx ; (198) let us suppose the integral to be definite, and the limiting values to be (x, y.), (Xn, Yx); it will be convenient in some cases to replace one or the other of these sets by general symbols x,y: also let us suppose an integral of (198) to be y=F(x); so that dy=f'(x) dx; then f(x) is subject to these conditions, yo = f(xo), (x) = f {x, F(x)}; and also to similar conditions at the superior limit. Let us suppose the interval X ,—X, to be divided into n infinitesimal parts, and X,, & ,, ... Xn-1 to be the values of x at the n-1 points of partition ; let the corresponding values of y be Yv, Y2, ... Yn-1; and let f (x,y) be finite and continuous for all these values : then Yı-y. = f (xo, Yo) (x1 — Xo), Yn-Yn-1 = f (x,-1, Yn-1) (@,, -Xn-1); and consequently Yo-Yo = (x,X.)f {x+0 (X, — Xo), %+ 0 (yu-yo)}; (199) where 0 is a general symbol for a positive proper fraction. Let us also express the greatest of the values of f(x, y.), f (x1, yı), ... f (x wyn) by A; then (199) takes the form, Yn-yo = (x,- X)0A; and thus (199) becomes Yn-y. = (x1 - x.)f {X, +0(X, — Xo), y + 01 (X— X,)}: (200) whence we know yn in terms of yo; and if for X, and y. the general values of x and y are substituted, we have y = %. + (x – X,)f{x+0 (x — xo), yo+01 (x − x)}. (201) It may be shewn, by a method similar to that employed in Art. 6, that the truth of (201) does not depend on the particular mode of partition of the interval x —-&o, provided that the parts of it are infinitesimal. 403.] There is however a condition to which (201) must be subject : we have supposed x, to be a constant; but as the differential equation does not assign any values at either limit to the variables, yo, although particular, must be arbitrary; and as y, and y must be continuous variables, one may be considered to |