: The total value of the integral is then less than the sum of the differentials d (sec' a), taken from x = 0 up to x, and it is greater than this sum taken negatively for in the first case we replace the variable factor cos 2mx by the constant quantity 1, and in the second case we replace this factor by 1: now the sum of the differentials d (sec" x), or which is the same thing, the integral fd (sec" x), taken from x = 0, is sec" x - sec" 0; sec" x is a certain function of x, and sec" 0 is the value of this function, taken on the supposition that the arc x is nothing. The integral required is therefore included between +(sec"x-sec" 0) and (sec" x - sec′′ 0); - that is to say, representing by k an unknown fraction positive or negative, we have always [{d (sec" x) cos 2mx} = k (sec′′ x − sec′′ 0). - sum of all the last terms of the infinite series. 187. If we had investigated two terms only we should have had the equation From this it follows that we can develope the value of y in as many terms as we wish, and express exactly the remainder of the series; we thus find the set of equations The number k which enters into these equations is not the same for all, and it represents in each one a certain quantity which is always included between 1 and -1; m is equal to the number of terms of the series 188. These equations could be employed if the number m were given, and however great that number might be, we could determine as exactly as we pleased the variable part of the value of y. If the number m be infinite, as is supposed, we consider the first equation only; and it is evident that the two terms. which follow the constant become smaller and smaller; so that the exact value of 2y is in this case the constant c; this constant is determined by assuming x = 0 in the value of y, whence we conclude It is easy to see now that the result necessarily holds if the arc x is less than 7. value X as near to In fact, attributing to this arc a definite as we please, we can always give to m k a value so great, that the term (sec x-sec0), which completes 2m the series, becomes less than any quantity whatever; but the exactness of this conclusion is based on the fact that the term sec x acquires no value which exceeds all possible limits, whence it follows that the same reasoning cannot apply to the case in which the arc x is not less than π. The same analysis could be applied to the series which express the values of 1, log cos x, and by this means we can assign the limits between which the variable must be included, in order that the result of analysis may be free from all uncertainty; moreover, the same problems may be treated otherwise by a method founded on other principles'. 189. The expression of the law of fixed temperatures in a solid plate supposed the knowledge of the equation 1 Cf. De Morgan's Diff. and Int. Calculus, pp. 605-603. [A. F.] A simpler method of obtaining this equation is as follows: If the sum of two arcs is equal to π, a quarter of the circumference, the product of their tangent is 1; we have therefore in general the symbol arc tan u denotes the length of the arc whose tangent is u, and the series which gives the value of that arc is well known; whence we have the following result: If now we write eV-1 instead of u in equation (e), and in equation (d), we shall have The series of equation (d) is always divergent, and that of equation () (Art. 180) is always convergent; its value is or - 1π. SECTION IV. General solution. 190. We can now form the complete solution of the problem which we have proposed; for the coefficients of equation (b) (Art. 169) being determined, nothing remains but to substitute them, and we have This value of satisfies the equation dev d2v =0; it becomes nothing when we give to y a value equal to or -; lastly, it is equal to unity when x is nothing and y is included between − 1 and +. Thus all the physical conditions of the problem are exactly fulfilled, and it is certain that, if we give to each point of the plate the temperature which equation (a). determines, and if the base A be maintained at the same time at the temperature 1, and the infinite edges B and C at the temperature 0, it would be impossible for any change to occur in the system of temperatures. 191. The second member of equation (2) having the form of an exceedingly convergent series, it is always easy to determine numerically the temperature of a point whose co-ordinates x and y are known. The solution gives rise to various results which it is necessary to remark, since they belong also to the general theory. If the point m, whose fixed temperature is considered, is very distant from the origin A, the value of the second member of the equation (a) will be very nearly equal to e cos y; it reduces to this term if x is infinite. The equation v= 4 ecos y represents also a state of the solid which would be preserved without any change, if it were once formed; the same would be the case with the state repre 4 -3x 3π sented by the equation v= term of the series corresponds to a particular state which enjoys the same property. All these partial systems exist at once in that which equation (2) represents; they are superposed, and the movement of heat takes place with respect to each of them as if it alone existed. In the state which corresponds to any one of these terms, the fixed temperatures of the points of the base A differ from one point to another, and this is the only condition of the problem which is not fulfilled; but the general state which results from the sum of all the terms satisfies this special condition. e cos 3y, and in general each According as the point whose temperature is considered is more distant from the origin, the movement of heat is less complex: for if the distance is sufficiently great, each term of the series is very small with respect to that which precedes it, so that the state of the heated plate is sensibly represented by the first three terms, or by the first two, or by the first only, for those parts of the plate which are more and more distant from the origin. The curved surface whose vertical ordinate measures the fixed temperature v, is formed by adding the ordinates of a multitude of particular surfaces whose equations are The first of these coincides with the general surface when x is infinite, and they have a common asymptotic sheet. If the difference v-v, of their ordinates is considered to be the ordinate of a curved surface, this surface will coincide, when a is infinite, with that whose equation is Tv, -e cos 3y. All the other terms of the series produce similar results. -3.r The same results would again be found if the section at the origin, instead of being bounded as in the actual hypothesis by a straight line parallel to the axis of y, had any figure whatever formed of two symmetrical parts. It is evident therefore that the particular values ce cos 5y, &c., ae* cos y, be** cos 3y, have their origin in the physical problem itself, and have a necessary relation to the phenomena of heat. Each of them expresses a simple mode according to which heat is established and propagated in a rectangular plate, whose infinite sides retain a constant temperature. The general system of temperatures is compounded always of a multitude of simple systems, and the expression for their sum has nothing arbitrary but the coefficients a, b, c, d, &c. 192. Equation (a) may be employed to determine all the circumstances of the permanent movement of heat in a rectangular plate heated at its origin. If it be asked, for example, what is the expenditure of the source of heat, that is to say, |