of certain differential Expressions , &c. 261 
= (■ '-£? -) +2W2 — g(l+CT)'F+F, (V) 
which, in fact, is Landen’s theorem; iorfdx ^ represents 
the arc of an hyperbola, semiaxes 1 and \/(e 2 — 1), and 'F, F, 
the arcs of two ellipses. 
In an analytical point of view, the latter part of this solution is 
unnecessary; for the problem is completely resolved, when it is 
proved that 
—m z )dz / I i — m z z? \ rJ / 1 1 — m z z z \ . 
V(-7=^)- /*V(- T=?) + 
If the differential of x J ( taken, it appears that 
and hence may be deduced a differential equation of the second 
order, similar to the one given in page 236. For, since — 
J » making e only to vary, or taking the partial differen- 
tials. 
d z f —ex z 
clx.de y[i—x z )(i—e z x z ) 
dx 
.and ,2 
dx dx.de ^(i— .r a ) (i_g 2 x z ) •> 
e ' de * de z » 
and 177 r-rr =/ — e 
J V(i— **) (i-e **) 7 d& 
Similarly, — 
J J <J[\ — x z ) (i — e a **)f 
or 7^-7rr^y (t=S?-) 
or irf + r--f- + (tz^) = o. 
when a; == 3, 
•/(0 + * 
<?/!■) I «•.■>*/(■) _ 
de 
O. 
I now purpose to show that the integration of forms such as 
