Remarks on Several Subjects. 137 
1 
T= —2sin. Yaay- *, sin. y, whose differential, put 
ry: we »?P 
=o gives, 2 y=ta ng. y 
Problem il. Ina Catenarian curve, in which a is the 
parameter, « es eescise, y the ordinate, and z the arch; 
; y=a. hyp. oe <a 
included by the pr and Gibhoriedntal: fine joining its ex- 
we have d=—~ =» and the area 
tremities u= 2(a+x) y— —2az. Ifwe suppose this to be 
maximum when Z is given 01 v=2, we shall have, by put- 
ting c=2l, a=z. ae 2 (a+c) 2th, u=2?. 
een > as , and v=z, hence m=Q, 
' am 
n=1. T (+ iid o7f - hyp. log. it Bat hose 
: ke ota +4it,) 
1a 
whence we may find ¢t and then a, a, z. This may be re- 
duced to the same form as page oso solution, by 
2 
differential put =o & reduced giveshyp. ie ye 
putting s=! — which gives log. <— =a: te Jog. Qt! 
2 log. ae and the wnsiiaes expression may be put 
under the form > —! hyp. log. “Fh =4, as in his solu- 
tio ‘ 
We shall add the i problem, not embraced in 
Professor Fisher’s ru 
Problem. Sepia idea y acy and *=2'y, a being a 
given numerical coefficient, and let it be required to find 
the value of ua maximum, when v is a constant quantity. 
The value of u is not homogeneus in z, ¥, but it may be 
made so by substituting Y* for y and then proceeding as 
above, or we ene at once put y=x7t, by which means we 
get u=x5(1+at), v=ax't, whence m=3, n=6, a 
Batt tba ye whose differential put =o ae 
nl 
1 
t=~_, whence y=2?t= 2, on c=V7 gy ay’ 
a 
Vou. VIII.—No. 1. 1s 
