1889. ] Equations with Boundary Conditions. 357 
this equation becomes y,dy,/v=y,dy,/w. And by combining the 
equations defining » and w with the equation connecting y, with 
y,, we easily deduce 
Cis (y.” a a’) ar {(a“y,” Pz: c) (yy 7, a)}? 
and Ws (Ys in a’) £ {(a*y," fe c’) (Ys a ae 
If we adopt the upper sign in the expressions for v and w, we 
obtain 
enter Os ee 
YW (yy i a’) ay ((a’y,’ ay c’) (y, sy a®)}? 
be YalYs 
ys (ys — 0) + (aye —c*) (2-0)? 
a eda, 
7 an, Hie Lf {(a*e,’ iV c’) (x, i ay} 
eda, 
dao 0, (eof =e) (02 — a 
Similarly if we started from the second and third of our original 
equations we should obtain 
Lae, 
w, («2 — a) + {(@x2 — 4) (a2 — a) 
i L0G, 
a, (2,2 — a*) + {(aa,? — c') (a? — a)? 
nf ody, 
ay? —c —y, (wy, —c’) (yZ - a)? 
edy, 
= ay,” =o = Ys (ays. im c’) Gi 71. a®)\> 
Multiplying these two sets of relations together, and taking 
the square root of the result, we obtain 
4 
y dy, 
; ; 7 
WY. (2a° je Y, ) Zs, cy? {(a’y,” a c’) (yy al a’)}* 
1 
Ys TY 
(Ys (2a° 7 Ys ) =¥ oy? ((a*y,. 7 c’) (Ys, EA a*)}4 
a2da, 
{a,? (2a” — x,”) — c'}2 {(ata,? — c*) (@2—a’)}4 
a a, da, te 
{a,? (20° — 7,2) — o'}4 {(a'a,’ — c*) (a — a?) }# 
* In this case the choice of signs may be verified by a similar method to that 
applied to the case in the preceding Article. 
