In comments Aaron asked about an example of Kevin's construction.

In Kevin's comment the rational points come from the group law on the elliptic curve and $D$ is the lcm of the denominators.

An example with $10$ points will be with so large $m,n$ it will be practically unreadable on MO,
so here is magma online code and example with $6$ points.

Starting with the OP $m,n=1370,2210$ got a Weierstrass model of EC.

Found $4$ generators with mwrank and worked with the first generator $P$ (could have used some of the OP points instead of finding generators). 

$P$ and $2P$ gave two additional solutions to the OP and here is the result:

    D= 7061463496178454923796024335945506461398736716739985
    m= 68314045389769361590347449258049778039693974321579389848056766335112029004282345656156971656752753986308250
    n= 110200029424372473806326907197291977713666922080795950046865294598976338758732834963581684205418676138497250
    x1= 7061463496178454923796024335945506461398736716739985
    x2= 162413660412104463247308559726746648612170944485019655
    x3= 261274149358602832180452900429983739071753258519379445
    x4= 204782441389175192790084705742419687380563364785459565
    x5= 21151048023226761412428898157420887062515978056922795
    x6= 35141206529629391701793045456458600088792043478723117




[Magma online code](http://magma.maths.usyd.edu.au/calc/)

    m:=1370;
    n:=2210;
    aa<x,y,z>:=AffineSpace(Rationals(),3);
    C:=Curve(aa,[m-x^2-y^2,n-x^2-z^2]);
    P:=C!([1,37,47]);
    pc:=ProjectiveClosure(C);
    E,m1:=EllipticCurve(pc,pc!(P));
    m2:=Inverse(m1);
    aInvariants(E);
    Ep:=E!([-323231734744697104/27633477663066497041,585299700649024 /27633477663066497041]);
    m2(Ep);
    m2(2*Ep);