therefore Z ́= [(xdp+ydq) = px + qy − z by (50); z=px+qy-Z = xX+yY-Z. On examining the above equations we see that x, y, z, and X, Y, Z are reciprocally related. Writing, side by side, the equations which are conjugate to each other, we have Z=Xx + Yy - z, z=xX+yY-Z. We see too that the equations (49) which express one set of the relations suffice to convert any relation found by integration between X, Y, Z, where Z stands for (X, Y), into a corresponding relation between x, y, z. Ex. Given z=pq. Here the transformed equation is px+qy-z = xy, of which the integral is z = xy+xf(2). Hence ↓ (X, Y) = XY+Xƒ(), and we have to eliminate X, Y, between the equations Each particular form assigned to ƒ gives a distinct par Y Y ticular integral. If we assume f(x)= X x=Y+b, y=X+a, 2 = XY, from which, eliminating X and Y, we have z=(x− b) (y — a), and this is one form of the complete primitive assigned in Chap. XIV. Art. 7. We may observe that the elimination may be so effected as to lead to general primitives. 11. In equations of the second order we should have, in addition to the above transformations, to change in order to form the reciprocal equation. Then the second integral of either being found in the form z= (x, y), that of the other will be found as before by eliminating X and Y from (49). For since therefore dx = RdX+ SdY, dy = SdX+TdY, The extension of the theorem to higher orders involves no difficulty. 12. It is an immediate consequence of the above, that any equation of the form • (p, q) r + † (p, q) s + x (p, q) t = 0.......................(52) can be reduced to an equation of the form x (x, y) r — ↓ (x, y) s + p (x, y) t = 0.........(53), usually more convenient for solution. Legendre's solution of the equation (1 + q3) r − 2pqs+ (1 + p3) t = 0, by the aid of the above transformation, will be found in Lacroix (Tom. II. p. 623). The same transformation makes the solution of any equation of the form Rr + Ss + Tt = V (rt — s2) dependent on that of an equation of the form Rr+ Ss+ Tt = V, but with different coefficients. The subject of these transformations has been most fully treated by Prof. De Morgan (Cambridge Philosophical Transactions, Vol. VIII. p. 606). 13. Legendre also shews how, by a transformation formally resembling the above, to integrate the equation r = f(s, t). Assuming s and t as independent variables, and v = sx+ty-q as dependent variable, the equation is reduced to the form dr dr where S and Tare the values of and furnished by the given equation. Lacroix, Tom. II. p. 631. ds dt EXERCISES. 1. To what condition must u and v be subject, in order that uƒ(v) may be a first integral of an equation of the form Rr+Ss+ Tt = V? Integrate by Monge's method the following equations: 2. x3r + 2xys+ y3t = 0. 3. qr-2pqs+p3t = 0. 4. Integrate ps — qr = 0. 5. Integrate by Monge's method the equation. q (1+q) r− (p + q + 2pq) s + p (1+p) t = 0. 6. The solution of Ex. 3 may, by the law of reciprocity, be made to depend on that of Ex. 2. 7. Monge's method would not enable us to solve the 8. Deduce by Poisson's method a particular integral of (1+q2) r — 2pqs + (1 + p2) t = 0. 9. Shew that the equations rt — s2 = ƒ (p, q), and rt — s2 = {ƒ (x, y)} ̄1, are connected by the law of reciprocity. 4x 10. The solution of the equation r―t= (rt - s2) p+q be derived from that of the equation r-t+ may Art. 7. Ex. 4. CHAPTER XVI. SYMBOLICAL METHODS. 1. THE term symbolical is, by a restriction of its wider meaning, applied more peculiarly to those methods in Analysis in which operations, separated by a mental abstraction from the subjects upon which they are performed, are expressed by symbols in whose laws the laws of the operations themselves are represented. du d Thus is written symbolically in the form u, the symdx d bol denoting an operation of which u is the subject. In dx thus expressing an operation by a symbol, in studying the laws of that symbol, and in founding processes and methods upon those laws, we introduce no strange or novel principle of Language; for it is the very office of Language to express by symbols the procedure of Thought. and so on. It will be observed that the symbol precedes the Operations may be performed in succession. Thus denotes that we first perform on the subject u the operation |