TRANSACTIONS OF THE SECTIONS. 17 



also 



U(2-00o88+^) =U (5+I) = U^=l. 



Scries of unitates have remarkable properties which fit them for practical use in 

 the verification of tables &c. Recurrence is a general and most valuable rule vt'ith 

 all series of unitates, and singular sequences are common. The series (to the base 

 9) for the squares is 1, 4, 9, 7, 7, 9, 4, 1, 9 ; 1, 4, 9, &c. By means of the series 

 for negative powers the unitates that correspond to certain circulating decimals 



may be assigned; for instance, - = •142857, 14, &c. has its imitate (to the base 9) 



equal to 4, — has its unitate equal to 1, and so on. 



The expression (10 — b)t-\-u also furnishes the means of obtaining imitates to 

 bases greater than 10, such as 11, 12, 99, 999, &c. For instance, by operating with 

 alternate digits, Ug36053 = U33ll.3=14. 



If the unitates (to various bases) of a number be given, it is possible to find the 

 number ; if 8 be less than 10, the number of unitates required fur the purpose wiU 

 (at least) be equal to the number of digits in the desired number. For instance, 

 required the two-figure number whose imitate to the base 9 is 5 and to the base 

 10, 4 ; this is found, by comparing the imitates of two-figure numbers to the said 

 bases, to be 14. 



Checking calculations, verifying tables, and ascertaining remainders to divisors 

 are tlierefore accomplished with ease by means of unitation. 



GrENEBAL PhtSICS. 



On Hills and Dales. By J. Clerk Maxweli, LL.D., F.R.SS. L. 4" E. 



After defining level surfaces and contour-lines on the earth's surface, the author 

 showed that the only measiu-e of the height of a mountain which is mathematically 

 consistent with itself is foimd by considering the work done in ascending the 

 mountain from a standard station. 



By considering a level surface, such as that of the sea, which is supposed 

 gi-adually to rise by the addition of water from the level of the deepest sea- 

 hottom to the tops of the highest mountains, he showed that at first there is but 

 one wet region round the deepest bottom. Afterwards other wet regions appear at 

 other bottom points of the surface and continually enlarge. For every new wet 

 region there is a bottom ; and when two wet regions coalesce into oue there is a 

 point where the surface is level, but neither a top nor a bottom, and this may be 

 called a Bar. When a wet region, as the water rises, throws out arms and em- 

 braces within it a di-y region, there is another level point which may be called a 

 Pass. The wet region then becomes cyclic. When the water covers the top of the 

 island thus formed the wet region loses its cyclosis again, and at last, when aU the 

 tops are covered, the wet region extends over the whole globe. Heuqe the number 

 of mountain-tops is equal to the number of passes plus one, and the number of 

 bottoms is equal to the number of bars plus one. 



The author then considered lines of slope which are normal to the contour-lines. 

 In general a line of slope is terminated by a top on the one side and by a bottom on 

 the other. At a pass or a bar, however, there is a singularity. Two lines of slope can 

 be drawn through this stationary point ; one of these is terminated by two tops and 

 is a line of watershed, the other is terminated by two bottoms and is a line of 

 watercourse. The watershed intersects the watercourse at right angles. 



If we consider aU the watersheds which meet at the same mountain-top, each of 

 these will reach a pass or a bar. The watercourses, which also pass through these 

 points, form a closed boundary', which is that of the region occupied by all the lines 

 of slope which meet at the mountain -top. This region round the mountain is called 

 a Hill. 



1870. 2 



