I have now posted photographs from the Charles Bertram display at the Magic Circle Museum in the Centre for the Magical Arts in London.
I believe these are the same cups and balls that Bertram used for the photos in The Modern Conjuror
. When you see the way the ball on the left in the b&w halftoned photo looks, you will see how it could appear to be crocheted.
The photos are at http://www.cupsandballsmuseum.com/bertram.htm
In The Modern Conjurer
, there are two photographs of the "fingertip palm" which is used to introduce a ball under a cup. One shows the hand with the ball at the tips of the little and ring finger. The other shows the ball being loaded under a cup. There are subtle differences between these two grips. Figure 3 shows the ball at the fingertips. Figure 4 shows the ball being introduced under the cup. Careful examination of the photo in Figure 3 indicates to me that the hand, as well as the ball, were lying on a flat surface, because there is no way to hold onto the ball with the fingers in that position, unless either the fingers or the ball are slightly tacky. Figure 4 gives a different finger position. In this picture the ball is clearly between the tips of those fingers, which will provide a secure grip.
These are not extremely small balls. They are about 3/4 inch or slightly larger in diameter. Bertram had fairly large hands. This is apparent from a couple of the photographs in the book.