SECT. I.] ANY INITIAL DISTRIBUTION. 345 



We can also write 



f+ao r+oo ? f**^ 



\ dj. &amp;lt;f&amp;gt; (a) sin qy. instead of I dif(o.} ee*s qx, 



J -oo J -oo 



for we evidently have 



0= [ &quot;diFtynnqx, 



J oo 



We conclude from this 



Too / r+oo 



TTV = e~ ht \ dq Q-W-t I da. $ (a) cos qy. cos qx 



JO V J - oo 



+ 1 da&amp;lt; (ajsin^sinja;) , 



J -00 / 



/oo /+ 



or, 7rv = e- M l dqe~ k&amp;lt; * H dx (a) cos ^ ( a), 



JO J -oo 



r + oo Too 



or, 7rv=e~ ht \ dz&amp;lt;l&amp;gt;(oL) I dqe- k * 2t cosq (x a). 



J -oo Jo 



355. The solution of this second problem indicates clearly 

 what the relation is between the definite integrals which we have 

 just employed, and the results of the analysis which we have 

 applied to solids of a definite form. When, in the convergent 

 series which this analysis furnishes, we give to the quantities 

 which denote the dimensions infinite values ; each of the 

 terms becomes infinitely small, and the sum of the series is 

 nothing but an integral. We might pass directly in the same 

 manner and without any physical considerations from the different 

 trigonometrical series which we have employed in Chapter ill. to 

 definite integrals ; it will be sufficient to give some examples of 

 these transformations in which the results are remarkable. 



356. In the equation 



7 TT = sin u + ^ sin 3z* + ~ sin ou + &c. 

 4 3 o 



/yi 



we shall write instead of u the quantity - ; x is a new variable, 



and n is an infinite number equal to -=- ; q is a quantity formed by 

 the successive addition of infinitely small parts equal to dq. We 



