Multiply by x, y, z, and add, then by (9) and (10), This is the equation to the surface of a wave of light propagated through a crystalline medium. See Fresnel, Mémoires de l'Institut, vol. vII. p. 136; Ampère, Annales de Chimie et de Physique, vol. xXXIX. p. 113; and Smith, Cambridge Transactions, vol. VI. p. 85. CHAPTER XII. ON PARTIAL DIFFERENTIAL EQUATIONS TO FAMILIES OF SURFACES. (246) We have seen in the preceding Chapters that families of surfaces might be expressed by means of equations involving arbitrary functions, on the form of which depends any particular individual surface of the family. We might eliminate by differentiation the arbitrary functions from these equations, and thus obtain partial differential equations to the families of surfaces: but it is equally possible to obtain these directly from the equations to the generator, as we proceed to shew. Let the equations to the generator be f(x, y, z, a, b, c...) = 0, f(x, y, z, a, b, c...) = 0...(1); in which a, b, c, ...are n parameters, connected by n 1 equations of condition; so that there is only one really independent, of which the others may be considered as functions. Now, to begin, let there be only two parameters, a and b, of which b may be taken as a function of a: then, if we differentiate the two equations (1), first with regard to x, and next with regard to y, considering z, a, and b as functions of these variables, we obtain four additional equations, while we introduce three new quantities, viz. db da da We have therefore, on the whole, five quantities, which may be eliminated between the six equations consisting of (1) and their four differentials. It is obvious that the result of the elimination must be a partial differential equation of the first order, since we proceed only to one differentiation. If there be three parameters, a, b, c, on proceeding to the second differentials, we obtain twelve equations, but we have then to eliminate twelve quantities, viz. db de d2b d'c da da d'a d'a d'a a, b, C, " đủ đã để để để dy để dxdy dự which is in general impossible; we must therefore proceed to the third differentiation when we find twenty equations between which we have to eliminate eighteen quantities, and the result gives two differential equations of the third order. It is easy to find, in general, the order of differentiation to which we must proceed in order to eliminate m parameters. Let ʼn be the required order of differentiation; then the number of quantities in the series a, dra da da d'a d'a d'a .. dra dy" is } (n + 1) (n + 2), while the successive differentials of the m - 1 parameters b, c... with respect to a, together with the quantities themselves, give (m − 1) (n + 1) functions; so that the total number of quantities to be eliminated is ļ (n + 1) (n + 2) + (m − 1) (n + 1). On the other hand, the number of equations, including the original ones, together with their differentials up to the nth order inclusive, is (n + 1) (n + 2). In order, then, that elimination may be possible, we must have (n + 1) (n + 2) > ¦ (n + 1 ) (n + 2) + (m − 1) (n + 1), (247) If the equations to the generator be given in the explicit u = c, v = $ (c).... form (1), the partial differential equation to the family of surfaces is easily found. For, supposing the functional equation to be Now, if the curve (1) lie on the surface (2), the values of the differentials dx, dy, dz, derived from (1), must satisfy equation (3). But if for shortness we put Eliminating dx, dy, dz, between (3) and (4), we find .(4). and therefore the partial differential equation may be written also under the form Pp Qq = R. Cylindrical Surfaces. (248) The equations to the generator are, in this case, as the partial differential equation to cylindrical surfaces. This equation may be applied to find the conditions that the general equation of the second degree may represent a cylinder. The form of the general equation is Ax2+ By2+ Cz2+2A'yz+2B'zx+2C'xy + 2A ̋x+2B'y + 2 C ̋z+ E=0, and we deduce from the preceding equation l(Ax+B′z+C'y+A′′)+m(By+C′x+A'z+B')+n(Cz+A'y+B'x+C')=0. Now, so long as the coefficients of x, y, z, in the latter of these two equations, are supposed to be finite, it is evident that it cannot hold good for all the values of the three variables which satisfy the former: we must have, then, since the coexistence of the two equations for all such values of the variables is required by the nature of the case, These are four relations between only two independent quantities (since the variables are, in fact, any two of the ratios 1:m:n); and therefore, in order that they may coexist, there must be two equations of condition between the constants. These are easily found by eliminating l, m, n, between the first three, and between the last and the first two, and the results are AA" + BB'2 + CC12 – ABC – 2A'B'C' = 0, A′′ (A'C' – BB′) + B′ (B'C' – AA') + C′′ (AB – C′2) = 0. Conical Surfaces. (249) The equations to the generator may be written and, dividing each member of the latter equations by the corresponding one of the former, by means of which equations, eliminating the differentials from |