CHAPTER IV. THE TREATMENT OF QUASI-CONSTANTS. 19. WE propose in this Chapter to develope more fully the doctrine of quasi-constants, and we think that it will be seen that by a right use of them a very great extension of integration-power is effected; and that this increase of power is almost wholly the consequence of the use of symbols of operation instead of brackets (Art. 4). From what has been said in Arts. 1 and 2, about the nature of quasi-constants, it will be understood that when we meet with such an equation as Dudu + Ed ̧u+nd ̧u + ¿d ̧u + ... we may treat it as an equation with constant coefficients. Now, on comparing it with the general formula of Art. 7, we find D1x=§, D1y=n, D1z = {, &c. and these by integration give (1), x = {t+,§, y = nt +$ ̧n, z = (t + $ ̧5, &c................. (2). ...... We might give to these integrals an apparently still more general form by writing 1 (§, n, 5.....), 4, (§, n, 5.....), P、(§, n, 5.....), ... for 6 ̧§‚, 6,1, 4,5, ... The forms above given are in general better suited for our purpose; and our conclusion from them is this, that if the independent variables of a new set are such that each one (as §) will satisfy an equation of the typical form x = §t +$§, then we know that and D1 = dt + dx + nd1 + Ed„ + ..., .. also F(D) = F (d2+ §d2+nd1 + çd1 + ...) (3). ...... 20. The simplest case of quasi-constants is found by equating each of the functions 1, 2,... to zero, in which case we have as the simplest set of quasi-constants that can be found. ... 21. If in the equation D1 = d + Ld2 + Md, + Nd2+ some of the coefficients L, M, N, ... are absolute constants, and some of them quasi-constants, we shall have F(D)= F(d+Ld + Md, + Nd. + ...) ....................... (5). ......... 22. We may here make one general remark which will apply to great numbers of equations. If we have an equation involving differential symbols of operation of one kind only, as de, or Dt, or A, ... then any absolute constant in that equation may be replaced by a quasi-constant, without in any way affecting the validity of the integral. We shall illustrate this property of quasi-constants by an example or two; and afterwards it will be assumed, even when not mentioned, that the compass of every differential equation and its integral may be extended by means of this substitution of quasi-constants for absolute constants. 23. We may also mention, for form's sake, what is obvious enough, that in integrating an equation in which two or more differential symbols of different kinds occur, the quasi-con stants corresponding to one symbol will not all (though some of them may) be quasi-constants under the other differential symbols. And, moreover, when the operating symbol in a differential equation is d, we may use functions of (d2, dy, d........) as quasi-constants. Ex. 1. To integrate tdu + xdu + yd,u =x"y'u°. Dividing by t we see that the coefficients of du and du are quasi-constants (Art. 20), and that § ==; y n = t Now this integral has been found on the supposition that a, b, c are absolute constants, but every step of the investigation will be equally valid if we suppose a, b, c to be functions Ex. 2. To integrate t'du + 2txdd ̧u + x2d ̧2u = tax". Dividing by t we find D=d+.d, and consequently Ꮳ the coefficients are quasi-constants, and §=2. #du + 2txd ̧à ̧u + xadu+b (ht+kx) (du+% d ̧u) + du+cu = 0. + Here as in the last example D=d+d, and §= .. t3Diu+bt (h + kğ) D1u + cu= 0, the integral of which is obtained as in ordinary integration for one independent variable; m and n being the two roots of the equation We are at liberty to write for b and c any functions of 2. Ex. 4. To integrate diu+2add ̧u+ad ̧u+ (du+ad ̧u) + cu = 0. b t си t2 t In this example D1=d+ad ̧; :. D1x=a, and x = at +§. In this example we are at liberty to write for b and c any functions of x at. Ex. 5. To integrate (ht + kx)2 (d3u + 2ad¿d ̧u + a3d ̧3u) Here we also +b (ht+kx) (du + ad ̧u) + cu = 0. assume D, dad, whence x at; 1 Dt (ht + kx) = h+ka; :. (ht + kx)2 Di̟u +b (ht + kx) D ̧u+cu=0. This is an equation of one independent variable, because D1x = a; .. u = (ht + kx)m. F§ + (ht+kx)”. ƒ§ = (ht + kx)". F(x − at) + (ht + kx)".ƒ (x — at), m, n being the values of m in the equation (h + ak)2 m (m − 1) + b (h + ak) m+c=0. We might add here the same remark as before respecting the constants b, c. 24. The preceding examples will also establish the principle, that if any equation of one independent variable, $(d) u = 0, can be integrated, then the corresponding equa |