Try changing
echo $this->Html->script('http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min');
to
echo $this->Html->script('http://ajax.googleapis.com/ajax/libs/jquery/1/jquery.min.js');
http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min doesn't seem to be a valid link.
It also seems that you need to wrap your carousel inner inside a div and then initialize it.
Changes to your html:
<div id="this-carousel-id" class="carousel slide">
<div class="carousel-inner">
.
.
.
</div>
</div>
And add this js:
$(document).ready(function(){
$('.carousel').carousel();
});