282 LAjVIBERT— ON THE SOLUTION OF [April 20, 



sin (;//) -I- cos nt — j- 



--^sin(;//) ^-, J, 



(9) ^W = A log /, F= ^ l^log / -f ^ ^^ - ^~- -^ 



2 X^ 2 • 3 -tr^ ~| 



"^ i^3 6T ~ KU' sT ^ ■ ■ ■ J ' 



(10) d{i) = At', V= Ac-^i + I -2T + f^ fl + • • •]• 



It will be noticed that in these solutions V =^0{t) is the heat dis- 

 tribution when ^■ = o. 



Solutions of the differential equation when 0(t) =0 correspond- 

 ing to several values of (j)(t) are as follows: 



III 0{t) = o, 



(i) <l>{t) = A, V=Ax, 



(2) (/>(/) = ^/, F=^[-r/+^y^], 



[2X^t 2X^ ~\ 



[,3 -5 



■*• + zh f T - ?hi' fy 



+ 2'K^'J\ J ' 



(5) <^(0 = At-K V= At~^ [r - ^^ ^ + ^ -^ 



3-5 x^ 1 



~ 2^¥ 71 +•••]' 



f— 3 2 ^T) ~l 



(6) <l>{t) = ^."'. F= Ae^" |^.r -H ^ ^', + ^,, ^', + • • • J . 



r « • S-1-' 



(7) <^[t) = /i sin (;//), V= A\ sin (;//).r -|- ^ cos (///) ^ , 



-^2Sin(«/)^, J, 



