SECT. II.] INTEGRAL FOR THREE DIMENSIONS. 375 



to be remarked since it is applicable to a great variety of cases ; ( 

 it is useful chiefly when the integral must satisfy conditions \ 

 relative to the surface. If we examine it attentively we perceive I 

 that the transformations which it requires are all indicated by f 

 the physical nature of the problem. We can also, in equation (j) t 

 change the variables. By taking 



we have, on multiplying the second member by a constant co 

 efficient A, 



v = 2 3 A fdnfdp fdq erW + * + f&amp;gt;f (x + 2n Jt, y + 2pji, z + 2$ Ji). 



Taking the three integrals between the limits oo and -f oo, 

 and making t = in order to ascertain the initial state, we find 



3 



v = 2 3 ^7r~2/(#, y, z). Thus, if we represent the known initial 

 temperatures by F (x, y, z), and give to the constant A the value 



-s _. 

 2 TT 2, we arrive at the integral 



8 r+ x r+*&amp;gt; r+ 



v = 7r~2 dn\ dpi 



J oo J oo J 



which is the same as that of Article 372. 



The integral of equation (A) may be put under several other 

 forms, from which that is to be chosen which suits best the 

 problem which it is proposed to solve. 



It must be observed in general, in these researches, that two 

 functions $ (as, y, z, t) are the same when they each satisfy the 

 differential equation (A), and when they are equal for a definite 

 value of the time. It follows from this principle that integrals, 

 which are reduced, when in them we make t = 0, to the same 

 arbitrary function F(x, y, z), all have the same degree of generality; 

 they are necessarily identical. 



The second member of the differential equation (a) was 



jr 



multiplied by ^ , and in equation (6) we supposed this coefficient 

 equal to unity. To restore this quantity, it is sufficient to write 



