Mr. HalHwellon the Boetianand Alabaldine Contractions. 13 



= r^ — / ,sm--{vt- Vr'^ + h^ + x'') 



X^ -y'- {r'^ + h^' + x^)^ ^ \ f 



which we see is a small quantity as long as a; is a quantity of 

 the same order of magnitude as X,, and h is many multiples of 

 r, which would still give the case of a point in the shadow of 

 a large circular disc, the aperture being an annulus; but to 

 take the extreme case which can be urged as an objection to 

 my former results, let h be small compared with ?•, or the 

 point deeply immersed in the shadow. The integral being 

 separated into two we have 



2'i^ax" P r .27r/, / \ 



/ — sin Ivt— V 7-2 + A2 , „2 I 



, 2'n^ax^ /* (h^-\-x'^)r . 2 tt / x 



We now reject the latter as very much smaller than the 

 former, and have only to perform the same integration as in 

 my former paper, and the intensity becomes for an aperture 

 which is an annulus and the radii i\ and r^, 



=4.a^\^(\- ^ysin^-J^^r^ + A^+^'-'/^-i^+A^ + ^A. 



The maximum intensity on the line being 4 a^ X^, there will 

 be a minimum equal to zero when .r = + — , and then the 



TT 



intensity will increase again, and apparently to a greater mag- 

 nitude than before, but our having used approximate expres- 

 sions prevents us pushing our conclusions further. 



We see clearly how maxima and minima are indicated by 



the theory in the plane perpendicular to the line in which I 



formerly discussed the results of the same method, and how 



much Mr. Tovey has mistaken the question. 



Queens' College, Cambridge, Dec. 8, 1840. RxCHARD PoTTER. 



III. T/ie Impossibility of the Boetian System of 'Numerical 

 Contractions, and the Alabaldine Notation having had a 

 common Origin. By J. O. Halliwell, Esq., F.R.S. 8fc. 

 C\^ a further examination of the Arundel MS. No. 343, 

 ^^ the real explanation of which has already appeared in 

 this Magazine, and on working out all the examples in that 

 manuscript with the tedious notation of the Boetian and 

 Alabaldine contractions, I have met with the following, 

 which shows more clearly than any other I have yet dis- 

 covered, the accuracy of the views taken by M. Chasles and 

 myself, on the impossibility of the Alabaldine system liavino- 

 commenced with the abacus, I merely give the early step^ 



