Motion of Viscous Liquids In Channels. 35 



-- = !+;, -1 + z, -l-i, 1-i: 



TT 



* = !J. (^W^¥T¥)^ { sinh {n+i)z+ sin ( ' l+i) * }' 



s 2 = - 2, t — r-rrs r-^ — r~i~- J cosh (n + l)-— cos (n + i)j I, 



tt^o (« + i)"cosh()i + ij7r I \ 2J j> 



(12) 

 (13) 



= ^ 2 „| fr + ti'U^ + i fr { Si " h (" + «*- Si " (" + «' }' (") 



* 47r4 =gl (n + i)4os 1 h ) (n + ^ { C ° 5h ^ + ^ + C0S f»+« ^ } > 



and so on. • • • (15) 



If in (12) we write z—%-\-\% and then take the real part, 

 f being real, we reproduce the expansion (11). 



Numerical Values. 



5. The series (6), (7), and (9) have been examined 



numerically by Mr. J. K. Maddrell, of Liverpool University, 



who has very kindly supplied the following results : — 



32 



? f " *' 



17*41374 



1 17-40685 



4 17-30824 



9 16-89301 



16 15-78149 



25 13-52752 



36 9-73117 



49 4-86775 



64 0-00000 



Here r and Xi refer to (6), while x and ^ 2 refer to (9), 

 results are shown graphically in figs. 4 and 5. 



The maximum value of % 2 is found to be about 1-1656, its 

 position being given by ,r/7r = "60819. This maximum has 

 an interest in connexion with several other physical problems 

 which are mentioned in the note referred to in § 3. 



The series for F/a 4 in (7) and (10) have the respective 

 values -9293, -02610. 



In connexion with a mathematically related problem Saint- 

 Venant * pointed out that if we write 



2F = kA 4 /I, 



whero A is the sectional area of the pipe formed by the sides 



* Comptes Rendus, t. Ixxxviii. pp. 142-147 (1879). 

 D 2 



8 





- X. 



TT 



X2- 



0.. 



. . o-oooooo 



1.. 



... 0-208361 



2.. 



... 0-546561 



3.. 



... 0-870728 



4.. 



... 1-096033 



5. 



... 1-163877 



6.. 



... 1-028783 



7... 



. . 0-650613 



8.. 



... 0-000000 



id V 



2 refer to (9). The 



