I've read your original request, the various responses that you received, and your follow up note.
At first, I thought your child was a victim of constipation. If that is the case, I recommend Omega-3 oil (in liquid not capsule form). My son has slow bowls and this helps a lot (it also tastes like lemons so it's not as bad as castor oil).
Your second note leads me to believe that it's not a biological thing, but rather a control issue on your daughter's part. We realized that my son (who was 2, 3, and 4 at the time) actually did hold his bowels. He did so, because it was painful and he was afraid of having bowel movements. When we got him to understand that it only hurts because he waited so long, things got a lot better for him. And now the omega-3's and lots of water keep him almost normal.
Now for your daughter, she's much older and at 10 she has a much broader understanding. Now, if you daughter has deeper psychological issues, I am going to apologize for what I am about to suggest. If she doesn't, you just might get her to start using the bathroom on her own. My suggestion: Let her wash her own poopy panties. She's 10. She's old enough and since she is choosing to do this (for whatever reason) she should deal with the consequences of her behavior. I feel confident (although I could be wrong) that cleaning poopy panties a couple of times will encourage her to get her butt to the bathroom when she needs to go.
Again, if you think I'm harsh, I apologize. But, I see your daughter as a full human being who is making a choice. It's her body, so you can't MAKE her do anything with it (the power is hers). But, you can make this a learning experience for her. Let her deal with the tangible effects of her behavior (poopy panties - the biological implications are meaningless to a 10 year old) and she may be motivated to change her behavior.
Hope this helps.
T.