The general solution of the equation is +(x,y)=Ÿ(x,y).¥{a(x,y), B(x,y)} (2) +(x,y)=ƒ(x,y).\{ƒ1(x,y), ƒ2(x,y)}, in which ƒ is a particular value of in the proposed equation, and f1s f2, particular solutions of (1). may The equation +(x,y)=(x,y).\{a(x,y), ß(x,y)} +Y1⁄2(x,y) be reduced to (2) by assuming ¥(x,y)=f(x,y)+P(x,y), in which ƒ is a particular value of y. (9.) Functional equations of the second and superior orders, containing two variables. Y2,1(x,y) = The equations 2,1(x,y) = a(x,y), may be treated in the same manner as the equations arbitrary functions of y being substituted for the arbitrary constants contained in the general solutions of the latter equations. A general solution of the equation (x,y)*= +(x,y) = =a is ax{(x − y)p(x,y), § (~~)} in which x, p, and §, are arbitrary functions. If is a homogeneous function of n dimensions, then \'+1(x,y)=√(x,y)|"". √(1,1)|i 1-n' -n (x,y) has the same signification with y(x,y), as used by Mr. Babbage. is The general solution of the equation ↓2(x,y)={\(x,y)} +(x,y)=Y{x(x,y) in which and x are homogeneous functions of n + 1, and n dimensions respectively, and such that p(1,1)=x(1,1). If the proposed equation is \"(x,y)=y{+(x,y)}, the general solution is √(x,y)=¿{P(x,y)} ̧ being as above, and § determined from the equation The same solution applies to the equation "(x,y)=y{y"-"(x,y)}, being in this case determined by the equation The equations .↓2(x,y)=a.↓21(x,y), y{\2(x,y),x}=y{a, y21(x,y)}, may be solved by the same assumption; ƒ being in the case determined by the equations (Babbage, Phil. Trans. 1816, pp. 184-222.) (10.) Given the equation &{a(x)}=d ̧¥(~), in which a2(x)=∞, then Given {a(a)}=d" f(x), and a'(x)=x; by successively substituting a(x), a2(x), &c. aox, for x, we obtain \(x)=dap-1(x) dap-2(x) • d'a(x) dr ↓ (x), a partial differential equation from the solution of which may be found. The equation 0= F{x,\(x),\{a(x)},.....\{a2 ̄1(x)}‚dTM1↓{a(x)},...dTMr-1\{a(x)} } in which a'(x)=x may be solved in a similar manner by substituting successively a(x), &c. a2-1(x), for x, and obtaining a series of equations from which all quantities except x, y(x), and its differential coefficients may be eliminated. (11.) Given (x,y)=d ̧¥{x,a(y)}, in which a2(y)=y, 1 then √(x,y) = €*., {y, a(y)} +e ̄* {a(y) — y} ·P2 {y, a(y)}, is the general solution, in which 1 and , are arbitrary functions. 1 If y(x,y) = d ̧{x, a(y)}, and a"(y)=y, then +(x,y)=d;↓(x,y), from the solution of which equation Given may be obtained. dy{x, ẞ(y)} = d ̧¥{a(x), y}, in which a2(x)=x, and ß2(y)=y; then may be found from the equation dẞ(y).d2 +(x,y)=d ̧a(x).d2√(x,y). The equation 0=F{x,y,\(x,y),.....\{a”(x), ß"(y)},.....d‡d;\{a′(x), ß*(y)} } in which a"(x)=x, and ẞ'(y)=y, may be solved in the same manner as the above equation of the same form, which contains only one unknown quantity. (Babbage, Phil. Trans. 1816, pp. 235—52.) APPENDIX. THE system of notation which has been adopted in the preceding pages, will have been found to vary in several particulars from that which has been long sanctioned and generally applied. It may therefore be expected that some sufficient reasons should be assigned for such a deviation from an established usage; and particularly as every change in the form of known symbols is calculated to interpose new difficulties to the pursuit of mathematical investigation. The opinions on the subject of notation which have induced the changes alluded to, cannot be more forcibly expressed than in the words of two of the ablest Mathematicians of the present day: "The great object of all notation is to convey to the mind, as speedily as possible, a complete idea of operations which are to be, or have been, executed; and since everything is to be exhibited to the eye, the more compact and condensed the symbols are, the more readily will they be caught, as it were, at a glance. * "The symmetry of mathematical notation (which should ever be guarded with a jealousy commensurate to its vital importance) facilitates the translation of an expression into common language at any stage of an operation, disburdens the memory of all the load of previous steps, and at the same time affords it a considerable assistance in retaining the results." * Edinburgh Encyclopedia, Art. Notation. |