140 
ME. L. X. G. FILOX OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
P = - 
2 L , r, L//' /cos S. cos cf).,' 
-log - — ■ -'-- 
TT 
Tcy TT 
'1 
'2 - 
SL - cos (2r + 2)4>,- cos (2v + 2) 4>, 
4L * rj-''+^ cos (2v + 1) cos (2v + 1) -rq- 
-(2PTT)T 
TT 0 
4L,/ ^ ,.^2-^+2 cos (2i. + 2) - ?-22''+- cos (2c A 21 0, 
0 /,2..+2 (2,, + 2) ; ^ 
V + S 
4Ly'« rj-‘'+’ cos (2c + 1) cos (2c + 1) </):, . _ ];:[ ) 
//-■■+>(2c+'!)! ^ 
^ L?/' /coS(^i cos(f)o\ 4L* 9 y‘'+^ cos( 2 c + 1 ) — ?y‘'+'cos ( 2 c + 1 ) tt 
^ ^ TT ^ rf/ ~ TT 7 + +'1): ■" 
I 4L//' “ ^ 1 “''+“ cos (2c + 2) 01 — r 3 -‘'+- cos (2c + 2) 0^ .p- 
^7^ 7 /F‘'+2 (2c + 2 ) ! 
4 L 7 / * 7 y.. + l (,Q 3 (2c+ 1) 01 —cos ( 2 c+ 1 ) 00 
irh 0 
P'+i ( 2 c + 1 ) 
(H,..i-TL.,). 
S = 
L , , , Ly' /sin 0 , sin 0 o'' 
l9i “ 9c) “ xr\ “ 
\ /I 
TT 
TT r-i 7’2 / 
4L - ri2''+2 sin (2c + 2) 0i - ^ 3 -*'+- sin (2c + 2) 0o 
— - X 
TT T 
j 2.,+2 ( 2 c y 2 ) ! 
4 L 7 /' * ^ 1 “''+’ sin r 2 c+ 1 ) 0 i—( 2 c+ 1 ) 00 
- IP.) 
_ V 
irli "0 
Z,2..+ i (2c + 1)! 
- H.,.) 
, 4 L 7 /' - 5 V "+2 sin ( 2 c + 2 ) 01 - /v "+2 sin ( 2 c + 2 ) 0 ., 
+ Trf- (2c a 2): --^=' 
The same remarks which were made on p. 106 as to the validity of such expressions 
apply here. Assuming that 2a' < Ah, we may apply these to obtain the state of 
things near the layer of shear and at its extremities. 
Clearly the only terms where discontinnities In U, V, P, Q, S, or their differential 
coefficients, may be introduced are their leading terms. Let ns therefore study 
these. 
It is easily seen that (a; -{-a') log ^i and {x — a ) log r.^ are finite, continuous, and one- 
valned throughout, tending to 0 at the points (T- 0). Their differential coefficients 
with regard to y' are likewise everywhere finite, but are indeterminate at (d- 0). 
They introduce, however, no discontinuity If we proceed along y = 0. 
Similarly y' log I’l and y' log are everywhere continuous, finite, and one-valued, 
and their dififerentlal coefficieuts with regard to give no discontinuity if we keep 
to y' =; 0, 
