the above expression reduces to x2 (q+n+2) Q, and vanishes if n is made equal to q-2. Thus (15) assumes the same form as if M and N were homogeneous of the degree p, and the condition of the theorem is satisfied. [If we write equation (21) in the form the required result follows at once from the remark on page 79, lines 4...7; for a homogeneous factor of the degree -q-2 will obviously render Qad integrable.] 7. All the applications which we have hitherto made of the partial differential equation (1) are of one kind. The general problem which they exemplify is the following. Under what condition does the equation Mdx+Ndy = 0 admit of being made integrable by a factor of the form (v) where v is a known and definite function of x and y? Let us examine the general form of its solution. On substituting (v) for μ in (1), we find μ The condition sought then is that the second member of this equation should be a function of v. Representing that function by ƒ (v) the corresponding value of μ is μ = € S1(v) dv μ .(23).. Any special case may be treated either independently as in the previous examples, or by directly referring it to the above general form. Thus a direct reference to the above theorem shews that the condition which must be satisfied in order that the equation Mdx+Ndy=0 may admit of an integrating factor of the form (x2+y) is that the function dM dN dy dx 2Nx-M should be a function of x2+y. And the mode of determining this point would be to assume a2+y=v, and, thence deducing y=v-x2, to substitute that value of y in the above function, and see whether the result assumed the form ƒ (v). The equation (23) would then give the value of μ. And this mode of procedure is general. 8. When by the discovery of an integrating factor the possibility of solving a differential equation has been established, there is no more valuable exercise than to endeavour to effect the same object by other means. Let us take as an example the equation considered in Art. 6, viz. 1 2 Pdx+P,dy+Q (xdy — ydx) = 0..................................(24), ......... P, and P, being homogeneous of the degree p, and Q homogeneous of the degree q. Let P ̧=x”4(2), P2=x3¥ (~), Q = x2 x (2) then the given equation, expressed in terms of the variables x and v, becomes x2þ (v) dx + x3¥ (v) (xdv+vdx) + x2x (v) × x2dv = 0, Now the reducibility of an equation of this form to a linear form has been established in Chap. II. Art. 11. Under the general form (24) are virtually included some remarkable equations which have been made the subjects of distinct investigations. Thus Jacobi has, by an analysis of a very peculiar character, solved the differential equation (Crelle's Journal, Vol. XXIV.)* (A+A'x+A"y) (xdy — ydx) — (B+ B'x + B'y) dy +(C + C'x +C"y) dx = 0.........(26). If, however, we assume in that equation x = &+a, y = n +ß, we can, by a proper determination of the constants a and ß, reduce it to the form (a§ + a'n) (§dn — nd§) — (b§ + b'n) dn + (c§ + c'n) d§ = 0, which falls under (24). On effecting the substitution in question the equations for determining a and B will be found to be a (A + A'a + A′′ß) − (B+ B'a + B′′ß) = 0, -B (A + A'a + A′′B) + C + C'a+ C"B=0. The most convenient mode of solving these equations is to write them in the symmetrical form B+B'a + B′′ß __ C+C'a+C"ß α = A + A'a+A”ß, then, equating each of these expressions to λ, we find A-λ+ A'a + A"ẞ = 0, B+ (B' −λ) α + B′′ß = 0, C + Ca + (C" − x) B = 0, *) Werke, Vol. 4, 52; J. 257-262 dip.305, Art.8 при projective transformation of the plane. from which eliminating a and ẞ we have the cubic equation (A − x) (B′ − x) (C′′ – λ) – B′′C′ (A — λ) — A′′ C (B' — λ) - A'B (C" -λ)+A'B" C+A"BC'=0............ (27). If a value of λ be found from this equation, any two equations of the preceding system will give a and B. 9. The present chapter would be incomplete without some notice of a method which was largely employed by Euler. That method consisted in assuming μ to be a function definite in form as respects the variable y, but involving unknown functions of x as the coefficients of the several powers of y. After the substitution of this form of μ in the partial differential equation (1), the result is arranged according to the powers of y, and the coefficients of those powers separately equated to 0. This gives a series of simultaneous differential equations for the determination of the unknown functions of x. But for the success of the method it is necessary that the primary assumption for μ should have been chosen with some special fitness to the object proposed. The following is an example. Required the conditions under which the equation Pydx + (y + Q) dy = 0 admits of being made integrable by a factor of the form P, Q, R and S being functions of x. In the partial differential equation (1), making clearing the result of fractions and arranging it according to the powers of y, we have (2p+dQ_dR 218333 dx dx ds dx y2 Whence, equating separately to 0 the coefficients of the different powers of y, we have the ternary system. The last equation gives ScQ, c being an arbitrary constant. Substituting this value of S in the equation obtained by eliminating P from the first two equations of the system, we find (2c-R) dQ+2 QdR = RdR, or, regarding therein R as the independent and Q as the dependent variable, and from the substitution of the value of Qin the first equation of the ternary system, |