146 
MR. L. X. G. FILOX OX AX APPROXIMATE SOLTITIOX FOR BEXDIXG A 
Gi (f) = (C„ + ;t)„) t 
we obtain the following homogeneous solutions in x, y. 
y 
. (109), 
Ur= 
V = 
X ofjb 
Sfjb (X + fx) 
+ :V 
Sfjb (X + fi) 
Ol/V 
— — (A„'iv_i - + -- (C„w„ + D„t'„) 
4/x 
ny 
P = A 
m 
a I 4 
(A,/y„ (A„?.f;,_i A B„'y„_j) ^ {(u,iVn — JI);i'?/-„) 
«(??. —1) \ -J3 /:hi —1) 
:5; yVn-2 ) H" B„( ^ v„_i -f -— 2/w«_2 
+ n ^, (110), 
A I (''- ~ 1) \ I -D — 1) 
Q — -^n ( ”1" y^7i—2 j ”l~ ( A "^n—l iy y^n~~2 
n (C„M„_i + D„t’„_i) 
o A I n(n, — l) \ (n n (n — 1) 
h = K \ — - - yUn - 2 ) + B,, “ Un-l -^- yVn--l 
4 
— n (U,— ]J,ytn-i) 
where u„, i’„ are the tAAn') homogeneous solutions of 3 -*^ + ^ = 0, thus, 
uX' ay" 
U,, = X“ — ^ 
n (n — 1) 
' y' + 
nx" ' y — 
n (n — 1) (n — 2) 
1 . 2 .^ 
x” ^ y^ 
+ 
and = 1, = 0, u_i = 0, v_^ = 0. 
We may add any number of such 2)olynoniial solutions. If Ave take n of them, 
beginning AAnth n = 1, and in the expressions ( 110 ) wnlte ^ = di we find (Q)+4, 
(Q)_i, (8)4,4 and (S)_4 each equal to algebraic polynomials in x of degree {n — l). 
A B 
Also, since A^, B,, I), come in only in the form I — Cj, y + D^, they are 
ecpuAadent to only tivo constants. We have therefore (4?i — 2) constants free. 
Noaa^ these are not enough to make Q and S coincide AA’ith any tAA’o giA^en 
})olynomials on the upper and loAver faces of the beam. Obviously, hoAA’OA'er, the term 
containing both in Q and S is independent of y and therefore cannot satisfy a 
perfectly general condition. If aa-c make this term disappear by Avriting C„ = A„/4, 
= — B„/4, AA^e haA’e iioaa' only 4n — 4 free constants left, but our polynomials 
