SECT. IV.] OTHER MODES OF INTEGRATION. 407 



following (c), (d), (e), to dynamical problems; the last (/) ex 

 presses what the movement of heat would be in solid bodies, if 

 the instantaneous transmission were not limited to an extremely 

 small distance. We have an example of this kind of problem in 

 the movement of luminous heat which penetrates diaphanous 

 media. 



404. We can obtain by different means the integrals of these 

 equations : we shall indicate in the first place that which results 

 from the use of the theorem enunciated in Art. 361, which we 

 now proceed to recal. 



If we consider the expression 



/+&amp;gt; /+&amp;lt; p 



dy. $ (a) I d&amp;lt; 



J - oo J -co 



cos (px-pz), .................. (a) 



we see that it represents a function of #; for the two definite 

 integrations with respect to a and p make these variables dis 

 appear, and a function of x remains. Thgjiataiir of the function 

 will evidently depend on that which we shall have chosen for 

 (j) (a). We may ask what the function &amp;lt;f) (a), ought to be, in order 

 tffSTafter two definite integrations we may obtain a given function 

 f(x^. In general the investigation of the integrals suitable for 

 the expression of different physical phenomena, is reducible to 

 problems similar to the preceding. The object of these problems 

 is to determine the arbitrary functions under the signs of the 

 definite integration, so that the result of this integration may be 

 a given function. It is easy to see, for example, that the general 

 integral of the equation 



dv d*v d 4 v d e v d*v - , 



would be known if, in the preceding expression (), we could 

 determine &amp;lt; (a), so that the result of the eq^kion might be a 

 given function f (x). In fact, we form directly a particular value 

 of v, expressed thus, 



v = e~ mt cospx, 



and we find this condition, 



m = op* -f lp* + rp 6 + &c. 



