SECT. IV.] FORM OF THE INTEGRAL FOR A RING. 397 



or the equation 



dv d?v 



To deduce from this differential equation the laws of the propa 

 gation of heat in a body of definite form, in a ring for example, 

 it was necessary to know the integral, and to obtain it under a 

 certain form suitable to the problem, a condition which could be 

 fulfilled by no other form. This integral was given for the first 

 time in our Memoir sent to the Institute of France on the 

 21st of December, 1807 (page 124, Art. 84) : it consists in the 

 following equation, which expresses the variable system of tem 

 peratures of a solid ring : 



/. 



(a). 



R is the radius of the mean circumference of the ring ; the integral 

 with respect to a. must be taken from a = to a. = ZnR, or, which 

 gives the same result, from a = irR to a = TrR ; i is any integer, 

 and the sum 2) must be taken from i = oo to i= + x ; v denotes 

 the temperature which would be observed after the lapse of a 

 time t, at each point of a section separated by the arc x from that 

 which is at the origin. We represent by v = F (x) the initial tem 

 perature at any point of the ring. We must give to i the succes 

 sive values 



0, +1, +2, +3, &c., and -1, -2, - 3 5 &c., 



and instead of cos write 

 M 



ix IOL . ix . la. 



We thus obtain all the terms of the value of v. Such is the 

 form under which the integral of equation (a) must be placed, in 

 order to express the variable movement of heat in a ring (Chap. IV., 

 Art. 241). We consider the case in which the form and extent of 

 the generating section of the ring are such, that the points of the 

 same section sustain temperatures sensibly equal. We suppose 

 also that no loss of heat occurs at the surface of the ring. 



