On the Canonical Proboscis

It is proved that the "canonical proboscis" domains corresponding to prescribed contact angle 7o, introduced in an earlier work by Fischer and Finn, are critical for the domain and angle, in the senses that (i) a solution of the capillary problem for angle -y in the absence of gravity exists over the domain if and only if -y is closer to 7r/2 than is 70, and (ii) singular behavior at y = 70 occurs precisely over the proboscis portion of the domain. The construction can be effected in a continuum of ways, allowing the proboscis to occupy as large a portion of the domain as desired.