is derived; here we consider the function u to be known, and we have ue-okt as the particular value of v. The state of the convex surface of the cylinder is subject to a condition expressed by the definite equation which must be satisfied when the radius x has its total value X; whence we obtain the definite equation thus the number g which enters into the particular value ue-okt is not arbitrary. The number must necessarily satisfy the preceding equation, which contains g and X. We shall prove that this equation in g in which h and X are given quantities has an infinite number of roots, and that all these roots are real. It follows that we can give to the variable v an infinity of particular values of the form ue-økt, which differ only by the exponent g. We can then compose a more general value, by adding all these particular values multiplied by arbitrary coefficients. This integral which serves to resolve the proposed equation in all its extent is given by the following equation v = a ̧u1e ̄økt + α‚μ‚ ̄økt +α ̧μ ̧ ̄Økst+ &c., 1] g? 91, 92, 93, &c. denote all the values of g which satisfy the definite equation; u,, u,, u,, &c. denote the values of u which correspond u12 to these different roots; a,, a, a,, &c. are arbitrary coefficients which can only be determined by the initial state of the solid. 307. We must now examine the nature of the definite equation which gives the values of g, and prove that all the roots of this equation are real, an investigation which requires attentive examination. In the series 1 gX + 2* 22. 42 22.42. 62 + &c., which expresses the value which u receives when x=X, we shall 9X2 replace by the quantity 0, and denoting this function of 0 22 by f(0) or y, we have y=f(0)=1–0+ the definite equation becomes Each value of furnishes a value for g, by means of the equation and we thus obtain the quantities g1, 92, 93, &c. which enter in infinite number into the solution required. The problem is then to prove that the equation must have all its roots real. equation f(0)=0 has all its We shall prove in fact that the roots real, that the same is the case consequently with the equation f'(0) = 0, and that it follows has also all its roots real, A representing the known number hX We write, as follows, this equation and all those which may be derived from it by differentiation, Now if we write in the following order the algebraic equation X=0, and all those which may be derived from it by differentiation, and if we suppose that every real root of any one of these equations on being substituted in that which precedes and in that which follows it gives two results of opposite sign; it is certain that the proposed equation X=0 has all its roots real, and that consequently the same is the case in all the subordinate equations These propositions are founded on the theory of algebraic equations, and have been proved long since. It is sufficient to prove that the equations fulfil the preceding condition. Now this follows from the general equation for if we give to a positive value which makes the fluxion +1 d'y di+1y vanish, the other two terms and de dei receive values of opposite sign. With respect to the negative values of 0 it is evident, from the nature of the function f(0), that no negative value substituted for can reduce to nothing, either that function, or any of the others which are derived from it by differentiation: for the substitution of any negative quantity gives the same sign to all the terms. Hence we are assured that the equation y = 0 has all its roots real and positive. 309. It follows from this that the equation f'(0) = 0 or y' = 0 also has all its roots real; which is a known consequence from the principles of algebra. Let us examine now what are the sucf'(0) cessive values which the term 0· or receives when we give f(0) y to values which continually increase from value of makes y' nothing, the quantity 1 = 0 to 0∞. If a = y becomes nothing y 3 Now it also; it becomes infinite when makes y nothing. follows from the theory of equations that in the case in question, every root of y' = 0 lies between two consecutive roots of y = 0, and reciprocally. Hence denoting by 0, and 0, two consecutive roots of the equation y' = 0, and by 0, that root of the equation y = 0 which lies between 0, and 0,, every value of included between 0, and 0, gives to y a sign different from that which the function y would receive if @ had a value included be 2 y 2 tween 0 and 0. Thus the quantity is nothing when 0=01; it 2 is infinite when 0=0, and nothing when 0=0,. The quantity o y must therefore necessarily take all possible values, from 0 to infinity, in the interval from 0 to 0,, and must also take all possible values of the opposite sign, from infinity to zero, in the interval from 0 to 0. Hence the equation A=0 necessarily has one y real root between 0, and 0, and since the equation y = 0 has all its roots real in infinite number, it follows that the equation A=0 y has the same property. In this manner we have achieved the proof that the definite equation + in which the unknown is g, has all its roots real and positive. We proceed to continue the investigation of the function u and of the differential equation which it satisfies. which serves to determine the coefficients of the different terms of the development of the function f (0), since these coefficients depend on the values which the differential coefficients receive when the variable in them is made to vanish. Supposing the first term to be known and to be equal to 1, we have the series x2 we make g 22 = 0, and seek for the new equation in u and 0, re garding u as a function of 0, we find |