37 
means that every X is teacher of every Y. This is a doubly 
total proposition (the preceding one is singly total), and 
such a proposition is inevitable, giving 
1 Y< B-nx 
A singly total proposition of the above form admits of the 
following four equivalent forms, whereof the truth or false- 
hood of each implies the truth or falsehood of all — Y signi- 
fying not = teacher, and X and Y not = X and not = Y : — 
1X< BY 
1V~ 1 1Y<X 
A' 1 1X< Y 
XF1Y< 0 
A doubly total proposition of the above form has four pairs 
of equivalent forms, as follows : — 
1X< A1Y 1Y< B-nX 
IVY<X_ _ 1F“ 1 X<Y_ 
1 Y< I" 1 F _1 X 1X< l- 1 FY 
XFY<0 Y F _1 X< 0 
The concluding part of the paper treats of the application 
of these principles to transitive relations, e.g. the relation of 
ancestor and descendant, where the ancestor of an ancestor 
is an ancestor. 
Joint Ordinary Meeting of the Society, and the Microscopi 
cal and Natural History Section of the Society, 
December 12th, 1882. 
H. E, Roscoe, Ph.D., LL.D., E.R.S., &c., President, in the 
Chair. 
Mr. James Heelis made some remarks upon the causes 
of the movement of the old Rhone Glacier with special 
reference to the power of gravity to produce such move- 
ment when considered in connection with the gradient 
down which the glacier has passed, 
