Z REPORT—1858. 
quite destitute of knowledge of their laws and progress; for when an unexpected 
stranger of this class blazes forth in our sky (as is the case at this moment), as soon 
as he has shown himself for a few days, we can mark the path which he will follow, 
the rate at which he will travel, and in a great degree the appearances which he will 
assume. And even objects which as yet are still more lawless and perplexing to our 
science than comets are, are still not altogether extraneous to the domain of our know- 
ledge. There is a class of such objects which has been especially attended to by the 
British Association, This is the subject of the first of the communications which are 
to be laid before this Section to-day. I speak of Prof. Powell’s ‘ Report on Luminous 
Meteors.’ These objects, falling stars, shooting stars, fiery globes, or whatever they 
may be commonly called, have attracted the attention of this Association for many 
years ; and the Report which we are to have laid before us to-day is the continuation 
of several Reports of the same kind prepared by the same gentleman in preceding 
years. These bodies, as I have said, are in a great degree irreducible to laws and 
extraneous to our science; yet not wholly so. We have speculations of recent times 
by some of our most eminent philosophers, in which these bodies play an important 
part. Prof. W. Thomson has been led, by his mathematical speculations on Heat, 
to the conclusion, that the heat of the sun is maintained by the perpetual falling in 
upon his surface of the abnormal bodies moving in the solar system, which appear 
to us as luminous meteors and shooting-stars. And he conceives that he has shown 
that there is in those bodies a sufficient supply to keep up the heat of the sun; and 
that, by the effects of them, the sun may have gone on radiating heat for thousands 
and thousands of years without the smallest diminution. And this, again, is the 
result of profound and complex mathematical calculations,—so wide is the domain of 
mathematical reasoning, and so necessary is it in any line of speculation in which 
we are to convert our ignorance into knowledge. I may mention, as another example 
of this, a case which is far removed from the vastness of astronomical phenomena, — 
a case of the manifestation of mathematical law upon a scale of the smallest dimen- 
sions, and in the work of a humble insect. I speak of the form of the cells of bees: 
a mathematical problem which already attracted the attention of the ancient Greeks, 
and which has been the subject of mathematical investigation by several of the most 
eminent mathematicians of modern times ;—the most eminent, for being a problem 
involving the properties of space of three dimensions, it requires‘considerable powers 
of mathematical conception. Upon this subject two communications are promised 
to the present Meeting, to be laid either before this Section or the Section of Natural 
History. And in order further to exemplify the advantages derived from the action 
of the British Association, I may mention another report upon a very different sub- 
ject, Mr. Cayley’s ‘ Report on the Progress of Theoretical Dynamics.’ The gene- 
rality, multiplicity, and complexity of the recent labours of analysts in this depart- 
ment of mathematics have been so great, that ordinary mathematicians cannot hope 
to follow them by reading the original memoirs; and I am greatly obliged, as one of 
them, to Mr. Cayley for enabling us compendiously and easily to understand what 
has been done and how it has been done. Perhaps, after all, his Report is not so 
very unlike that of Prof. Powell ‘On Luminous Meteors ;’ for the original researches 
of the great analysts who have treated this subject, though bright and objects of won- 
der, are so far above our head and so difficult to understand, that they are not unlike 
the things tabulated in the other Report. And now, having explained that we must 
often be necessarily difficult to follow in this Section, I must ask the ladies and gen- 
tlemen here present, as the Spectator asks his readers, to believe that, if at any time 
we are very dull, we have a design in it. 
On a General Method of deriving the Properties of umbilical surfaces of the 
second order, having three unequal axes, from the properties of the sphere. 
By the Rev. J. Bootn, LL.D., F.R.S. 
The author called the attention of the Section to the researches of M. Chasles and 
other French geometers, on the methods of deriving the properties of surfaces of re- 
volution from those of the sphere by the method of “ Reciprocal Polars,’’ and called 
attention to the fact, that they did not grapple with the more general problem when 
