SECT. IV.] ELEMENTS OF THE METHOD PURSUED. 451 



movement of heat in a sphere of given radius, is very different 

 from that which expresses the movement in a cylindrical body, or 

 even in a sphere whose radius is supposed infinite. Now each of 

 these integrals has a definite form which cannot be replaced by 

 another. It is necessary to make use of it, if we wish to ascertain 

 the distribution of heat in the body in question. In general, we 

 could not introduce any change in the form of our solutions, with 

 out making them lose their essential character, which is the repre 

 sentation of the phenomena. 



The different integrals might be derived from each other, 

 since they are co-extensive. But these transformations require 

 long calculations, and almost always suppose that the form of the 

 result is known in advance. We may consider in the first place, 

 bodies whose dimensions are finite, and pass from this problem to 

 that which relates to an unbounded solid. We can then substitute a 

 definite integral for the sum denoted by the symbol S. Thus it is 

 that equations (a) and (/8), referred to at the beginning of this 

 section, depend upon each other. The first becomes the second, 

 when we suppose the radius R infinite. Reciprocally we may 

 derive from the second equation (ft) the solutions relating to 

 bodies of limited dimensions. 



In general, we have sought to obtain each result by the shortest 

 way. The chief elements of the method we have followed are 

 these : 



1st. We consider at the same time the general condition given 

 by the partial differential equation, and all the special conditions 

 which determine the problem completely, and we proceed to form 

 the analytical expression which satisfies all these conditions. 



2nd. We first perceive that this expression contains an infinite 

 number of terms, into which unknown constants enter, or that 

 it is equal to an integral which includes one or more arbitrary 

 functions. In the first instance, that is to say, when the general 

 term is affected by the symbol S, we derive from the special con 

 ditions a definite transcendental equation, whose roots give the 

 values of an infinite number of constants. 



The second instance obtains when the general term becomes an 

 infinitely small quantity ; the sum of the series is then changed 

 into a definite integral. 



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