272 report— 1879. 



consequence of the remainders used being imitates, that is, simply the number of 

 units by which the number to be dealt with is in excess of being exactly divisible 

 by the divisor. This divisor may, in practice, be either 9 or 11. 



As an instance, suppose the calculation to be checked is w = 62 - 32, 2'375, 3*25, 

 3 - 75 = 1803-871875. In casting out the nines the symbolisation is U 9 w = U 9 

 (4.8.1.6) = 3. In casting out the nines there is no check upon the number of digits in 

 the number operated upon, neither is there a check upon the place of any particular 

 digit, nor upon the figures themselves, if they be either 9 or 0, or if their sum be 

 9 or any multiple of 9. In 100 there are 33 fractional imitates to reciprocals. 



In casting out the elevens there is a check upon the number of digits, upon the 

 place of any digit, and, for the most part, upon the figures themselves. In 100 

 there are only 9 fractional unitates to reciprocals. U» (\, %, £, \, |, £, I, ^, f , ~, ~) = 

 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, i 



Additions can be dealt with at one operation. Subtractions must have the 

 unitate of the minuend made greater than that of the subtrahend. Decimal 

 multiplications must be without contraction ; but divisions may be finished at any 

 predetermined place of decimals, taking into account the remainder. Fractions 



are treated as if of the form a.-^. 



The table of powers of TL,n repeats after every ten powers, and is therefore 

 very serviceable for checking tables and formulae in which the higher powers occur. 



An Appendix, containing examples, tables, and illustrations, accompanies the 

 original paper. 



MONDAY, AUGUST 25, 1879. 



The following Reports and Papers were read : 



1. Report of the Committee on Tidal Observations in the English Channel. 



See Reports, p. 71- 



2. Beport of the Committee on Calculations of Sun-heat Co-efficients. 



See Reports, p. 66. 



Report of the Committee on Luminous Meteors. 

 See Reports, p. 76. 



4. On the Direct Motion of Periodic Comets of Short Period. 

 By Professor H. A. Newton. 



In the ' American Journal of Science ' I published a few months since an article 

 on the origin of comets. I undertook in that article to find out if there is in any 

 facts we know about the comets reason to say whether they must have come to us 

 from outside space, or whether they have been formed out of matter that lay on 

 the outer edge of the disc-shaped nebula which the solar system is supposed to have 

 been condensed from. The comets may be divided into two very distinct classes ; 



