8 REPORT—1862. 
cients of the general term of the binomial theorem, as explained in the first memoir. 
In this the expansion was effected in terms of p and 7, but we may suppose the 
expansion effected in terms of (p) alone. In that case the coefficient of the general 
term would be symbolical, and a function of (7). He had calculated its value in the 
memoir, and also the value of the corresponding general symbolical coefficient in 
the multinomial theorem supposed expanded in powers of p alone. He concluded 
the paper by giving a method to expand the reciprocal binomial (7?+ 6 (p) dz)” in 
terms of (7). The general cases of division yet remained to be worked. This has 
been effected by Mr. Spottiswoode in a very able and beautiful paper published in 
the ‘Philosophical Transactions’ for 1862. He has there given in full the division of 
gn (p) a" + pn—1 (p) 7"! +hn—2(p) 7” 7 +.&C, «+ +, (p) 
internally and externally by W, (p) 7+, (p); secondly, the division of 
Pn (P)™ +Pn—1(p) + Pn—a (p) "bese + tho (P) 
internally and externally by 
Ym (p) + m_1 (p) + Yn—o (p): 7-3-4. os Wo (p) 5 
thirdly, the division of 
p” Pn (7) +p"—" Pay (m)+p"~" bn» (m)+. 3" +, (7) 
internally and externally by 
OPV (+P n—1 (7) +p” Wma (4) ++ +++ Yo (m7) 
He has fully investigated the conditions that the divisor in each case may be an 
internal or external factor of the dividend, and his results, which are expressed by 
means of determinants, will be found extremely interesting. The author in conclu- 
sion states that he believes the form in which the calculus now stands will be per- 
manent, and that subsequent improvements will be very much based on extending 
sees of multiplication and division to other symbolical expressions, in which 
the laws of symbolical combination are different from those here assumed, 
On some Models of Sections of Cubes. By C. M. Wrtttcn. 
These were carefully-executed models, designed to illustrate certain simple pro- 
positions in solid geometry relative to the volumes, &c. of solids formed by the 
section of a cube by planes. The author wishes, at the same time, to place on 
record the simple fraction 444, which gives an extremely close approximation to the 
side of a square equal in area to a circle of which the diameter is unity. 
ASTRONOMY. 
Some Cosmogonical Speculations. By Isaac Asun, VB. 
The author considered that the present planiform condition of the system dis- 
proved the common view that it had formerly been a gaseous sphere, and proved 
that it had originally been a liquid plane, as Batarnts rings are at present; nor yet 
in a heated condition, since he thought that, though capable of transformation, 
heat could no more be absolutely Jost than its equivalent, motion. The planets had, 
doubtless, been originally molten; but this heat the author ascribed to the collision 
of particles, during their formation, from the liquid plane described. This formation 
he ascribed to the development of a centre of attraction in the liquid plane, and 
showed how, in a revolving plane, a diurnal rotation from west to east might hence 
be originated, the particles so attracted acting as a mechanical “couple” of forces 
on the planet during its formation. From the distance between the interior and 
exterior planets, he inferred the former existence of two rings, as in the system of 
Saturn, the asteroids being probably formed from small independent portions of 
matter between these rings. He considered that the planets also first existed in- 
dividually as planes, basing this view on the uniformity of plane observed in the 
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