CONSTITTJTION OF MATTER AND ANALTTICAL THEORIES OF HEAT. 37 



Therefore, using (1) o£ the last article, it foUows from the end of Art. 33 



that 



Thus 



,|i3iirwi+2i&+i|-iM|. 



iJ<^A,{l3i|r(;r)l + 21& + i|-iM|. (m) 



36. Consider £. 



J{x,t) = f Y' {x, t')ät'='^^ sin mx \ e""''^ - 1 1 ; 



0 1 



hence J'(x,t) — '^a^cosmx\e~^ ^ — l] 

 1 



CO 



and, consequently, \J'{x,t)\:<L'^\a^\, 



1 



wliatever x and t may be. Let Jj stand for the greatest value that | J' (x, t) \ 

 can have, whatever x and t may be. 

 Then 



^:^SI«J<^|-If + (1) 



Now K\J{x,t)\ is evidently less than 2Kl^J^, iix does not lie in the 

 interval (— :r + 2X^, it — 

 Therefore 



(IV) 



3 



It at once follows, from the procedure in the case of W in Art. 33, that 



.<2A||f(^)| + |I^|. (V) 



Sufficient and quasi-necessary conditions for f{x). 

 37. It follows from (III) and (IV) that 



3.B+^<M|43i/■' (^)l + 676 + f|-lM|. (1) 



of g and g^, that 



— 1 1 / I 



Now, let f stand for — / f ix') äx' . Then it is evident, from the definitions 



g = '^Kitf, 



