NOTICES AND ABSTRACTS 
OF 
miscellaneous communications to the sections. 
MATHEMATICS AND PHYSICS. 
Mathematics. 
On the Solution of Cubic and Biquadratic Equations. 
By Arthur Cayley, F.R.S. 
On an Extension of Quaternions. 
By Sir W. R. Hamilton, LL.V., M.R.I.A. ions which 
i * le auth° r showed how, by a slight change in the formula: of sector an d 
vers ad * lread y published, and a slightly different interpretation ^ gp heroid. 
r . they became applicable to the investigation of the proper 
Moigno presented to the Section a new Ari&mom^'° he 
me by M. T. De Colmar; and as the Abb* spoke English ^.thdjfficu 
v A j r °f- Wilson to explain the machine to the Section. inches long 
by c :' :l 7 beautifully executed, consisted of an oblong box, n ’<> • ,| t0 turn 
4nd . wide - 0,1 the ^ce, the machine was furnished w >^Z etic scale, 
I. a number of small holes, at which the digits ot the c °? im k | nn d which 
fiUv 4 *’ 5 ’ I' 8 »9. made their appearance as the machine ’ but they 
ri "Rht t» RVe ftnswcr< In this machine they were eight in , ,» be 5 et to the 
rt1 aji,Mi e /*^ n ^ e< ^ ,0 an y number. To each of these was an 1 . - 1V0f y ball to 
^ 'noved'fln en l raved ou a 8muU attached vertical 5 cale,and a operation to be 
tonduetDrlK ?i K lts scale according to certain simple rules, a 1 y draw- 
i:,_. I he machine varied from addition to multiplication, d t0 view. 
Thi« thr.ii sluI,n S bottom of the machine, the machinery w ^ cb)nC 0 r dia- 
»r an j s R* 1 simple, could not be intelligibly explained without they 
•*«, "» rt of it —**4 •* 4* cylinder* » *™"Sf“ > SWon » 
hi- hal.l ; n , glt *' enamelled on a circle at their upper parts, c ® ■ m, r0 ., n d them 
m , { e ^ ace ’ w 111,6 by a number of indentations arTMiB® P ' 7 ( - ace a t the 
<■• 9 . " h ' cb ,lre iM >“ w« ~t would be stopped at “ “ e rf pin i„ns . 
r, 'nnexi 0n ^ C " ,rig to that at which the index was set ; whil y bftn |(.note ma¬ 
rine, t0 them something similar to that m the con .^-thus by 
1 854 . “ 1 addlt >on could bo performed and the result appear on the ^ 
