1889.] Equations with Boundary Conditions. 353 
i COO (ie ns On) (Oe) 
Also y,- (e, — hy = (a, —h)? ; 
and thus it follows that 
Thus we have the relations 
dy, .= dy, 2 Dipas! da, ag da, : 
TO 2 Oo ee 
and, when combined with the equations of Article 3, these give 
Ou Ou Ou Ow 
RB Gon Wes 2 2 $2 
(B= 8) = ye —8) 5 = 8) n= 8D 5 
We are therefore at liberty a assume that 
du =O au A 
oy y—& Seen oS 
all along the curve 
ay —h(a+y)+e&=0. 
We therefore obtain 
y— o +8) (h—8) 
25 {log SS 5 eas ~°°8 (a — 8) (h+ =f 
os ay —O(a@tyy+O | eee 
= 55 sae ee een ; 
and consequently 
k—6 h—6 
i oS - los at: 
These equations determine the solution of the problem, and we 
have for the solution of the original problem 
2 _— 
Se es 
Beto ee) tog FES 
S (b+ 8) (h —6) 
7. In the cases worked out in Articles 5 and 6, the expressions 
involved are made to fall very easily into the required forms. This 
VOL. VI. PT. VI. 26 
