Once the sugar is dissolved, reduce the heat to low so the syrup remains warm.
Fill clean, sterilized canning jars with apple quarters. Be sure to run your fingers around the tops of the jars before using them to check for tiny nicks. If you find any, those jars are no longer useful for canning.
Cover the apples with the water and sugar syrup you just made. Leave 1/2 inch air space at the top of the jar.
Wipe the top and edges of the jar with a clean, damp cloth.
Put a clean, sterilized, new lid on the jar, add the band and screw it down until it is fingertip tight.
Place the jars in a cold pack canner filled with boiling water.
Process both pints and quarts for 20 minutes.
Turn the heat off under the canning pot once the 20 minutes is up.
Remove the lid, but allow the jars to sit under the water for 5 minutes longer.
Carefully remove the jars. Set them aside and allow them to cool overnight.
Remove the bands and check the seals by gently prying them with your fingertips.
If the jars are sealed, replace the bands, label and store in a cool, dark, dry place.
If the jars are not sealed you can either select a new lid and attempt to seal the jars again or put the food into your refrigerator and use it up.