Transpinite Cardinal Numbers of Well-ordered Aggregates. 61 
average range of vertical disturbance being about 00140 c.a.s., 
differences of this order may be obtained by supposing that 
the slabs of magnetic matter are 16 kiloms. (10 miles) thick, 
and that the upper surfaces are 4 kiloms. from the surface of 
the earth. We may, however, arrive at a similar result by 
assuming that iron exists in the crust of the earth chiefly in 
the form of magnetite and iron ores, and that the tempe- 
rature at which these cease to be magnetic is 555° C. With 
a rise of temperature of 1° per 90 feet of depth this would 
give adepth of 15 kiloms, or 9$ miles, as the thickness of the 
magnetic floor, a value which corresponds with that calculated 
from a knowledge of the range of vertical disturbance. 
In conciuding this description of the experiments, which 
were carried out in the Physical Laboratory of Birmingham 
University, I take the opportunity of expressing my thanks 
to Prof. Poynting for suggesting the investigation, providing 
the necessary apparatus and space, and for much encourage- 
ment and assistance during the progress of the experiments. 
I am also indebted to all the members of the Geological 
Department of the same University for their cordial assist- 
ance with the geological side of the investigation. 
Physical Laboratory, 
Birmingham University, 
May, 1903. 

VIL. On the Transjfinite Cardinal Numbers of Well-ordered 
Aggregates. By Puuip HE. B. Jourpain, B.A., Trinity 
College, Cambridge *. 
N the memoir f in which the transfinite ordinal numbers 
first appeared in a form independent of the theory of 
the derivatives of aggregates of points representing ‘real 
numbers, the illustrious author, Georg Cantor, defined a 
series of “powers”’ which belong to various classes of the 
transfinite numbers, and which series possesses the remarkable 
property of being ordinally similar to the whole class of 
transfinite numbers f, 
* Communicated by the Author. 
+ “Ueber unendliche, lineare Punktmannichfaltigkeiten, v.,” Math. 
Ann. xxi. pp. 545-591 (1883) [dated December, 1882/; also as a separate 
pamphlet, ‘Grundlagen einer allgemeinen Mannichfaltigkeitslehre. Ein 
mathematisch-philosophischer Versuch in der Lehre des Unendlichen,’ 
Leipzig, 1883. Pagez of the “Grundlagen ” corresponds to page n +544 
of the article mentioned first. 
t ‘Grundlagen,’ pp. 43-44. 
