338 Simple Shearing Motion of Viscous Incompressible Fluid. 



The hyperbolic functions then disappear and the equation 

 reduces* to 



*i(%)-*s(«/«)— *«(%)• *i(%)=0, . '. . (40) 



which cannot be satisfied by a moderately large value of rj 2 . 

 For it appears from the appropriate expressions (21) . . . (24) 

 that the left-hand member of (40) is then 



DI y,2V2.„3/2 cos(7r/6)) 



a positive and rapidly increasing quantity. Again, it is 

 evident from Table I. that the left-hand member of (32) 

 remains positive for all values of ?? 2 from zero up to some 

 value which must exceed 1*1, since up to that point the 

 functions Sj, s 2 , t 2 are positive while t x is negative. Even 

 without further examination it seems fairly safe to conclude 

 that (32) cannot be satisfied by any values of r l2 an( i X. 



Another case admitting of simple treatment occurs when 

 7) 2 and 77! are both small, although X may be great. We 

 have approximately 



*i=l,. *i=--f?7 3 , s 2 = j?7 4 , h=V> 



the next terms being in each case of 6 higher degrees in rj. 

 Thus with omission of terms in n 7 under the integral sign, 

 (31) becomes 



(e Xv d V .( V e- Xr >d v - (e-^d v .(r,e Xri dri = 0, (41) 



or on effecting the integrations 



M^a— *7i) smn MV2— Vi) +2 — 2coshX(r7 2 — ??i) = 0. (42) 



It is easy to show that (42) cannot be satisfied. For, 

 writing \(r)„— 7j 1 ) = x i 



as sinh x = x 2 -J- -5— + 



2.3^2.3.4.5 ' "■' 



X* X 6 



2(cosh^-l) = ^ 2 +^-f 3>4>5j6 -t- •••> 



every term of the first series exceeding the corresponding 

 term of the second series. The left-hand member of (42) is 

 accordingly always positive. This disposes of the whole 

 question when rj 2 and 7] l are small enough (numerically) r 

 say distinctly less than unity. 



* Regard being paid to the character of the functions. Needless to 

 say, it is no general proposition that the value of an integral is deter- 

 mined by the greatest value, however excessive, of the integrand. 



