274 THEOKY OF HEAT. [CHAP. V. 



289. We now know a particular form which may be given to 

 the function v so as to satisfy the two conditions of the problem. 

 This solution is represented by the equation 



Ae~ knH sin nx , . sin nx 



v - or v 



, 2 . 



-Kn t 



x nx 



The coefficient a is any number whatever, and the number n is 



n X 

 such that - - Tr=lhX. It follows from this that if the 



initial temperatures of the different layers were proportional to 



the quotient - - , they would all diminish together, retaining 

 fix 



between themselves throughout the whole duration of the cooling 

 the ratios which had been set up ; and the temperature at each 

 point would decrease as the ordinate of a logarithmic curve whose 

 abscissa would denote the time passed. Suppose, then, the arc e 

 being divided into equal parts and taken as abscissa, we raise at 

 each point of division an ordinate equal to the ratio of the sine to 

 the arc. The system of ordinates will indicate the initial tem 

 peratures, which must be assigned to the different layers, from the 

 centre to the surface, the whole radius X being divided into equal 

 parts. The arc e which, on this construction, represents the 

 radius X, cannot be taken arbitrarily; it is necessary that the 

 arc and its tangent should be in a given ratio. As there are 

 an infinite number of arcs which satisfy this condition, we might 

 thus form an infinite number of systems of initial temperatures, 

 which could exist of themselves in the sphere, without the ratios 

 of the temperatures changing during the cooling. 



290. It remains only to form any initial state by means of 

 a certain number, or of an infinite number of partial states, each 

 of which represents one of the systems of temperatures which we 

 have recently considered, in which the ordinate varies with the 

 distance x, and is proportional to the quotient of the sine by the 

 arc. The general movement of heat in the interior of a sphere 

 will then be decomposed into so many particular movements, each 

 of which is accomplished freely, as if it alone existed. 



Denoting by n lt n a , n 3 , &c., the quantities which satisfy the 



equation - - ^=1 hX, and supposing them to be arranged in 



