16 REPORT 1870. 



On Linear Differential Equations. By W. H. L. Ettssell, F.B.S. 

 The object of this paper was to explain certain discoveries made by the author 

 in linear difl'erential equations, and chiefly to solve the general equation of the wth 

 order, whenever that solution isof the form y = P€", P and a being rational and 

 entire functions of (or). 



On a Numencal Theorem, with practical applications. 

 By W. H. Walenn. 



This novel and practical theorem is, " That if t be the tens', and u the units' 

 digit of a two-figure number, and 5 be any integer less than 10, then 



(10— 8)H «< 

 has the same remainder to 8 as 10<-l-««." 



For {\0—b)t-\-u, when expanded by multiplication, becomes \Ot—U-\-u, or 

 (lQt+u) — U; this latter expression only differs from ¥it-\-u by an exact number 

 of times 8, and therefore has the same remainder to 8 as 10<+m. 



When this theorem is adapted to other than two-figure numbei-s, the expression 

 (10 — S)<+2«, by expansion, becomes 



(10-8)»-ia+(10-8)''-'6+(10-8)''-3c-|- +(10-8)^s-f-(10-8)i!-f-«, 



if n= the number of digits or figures in the given number; for each time 10 

 occirrs as a factor in any term, it must be treated in the way above indicated. 



The remainder to any digit may be determined by means of the expression 

 (10 — 8)<+M without the knowledge of any multiple of that digit. When the 

 arithmetical operation indicated by the formula {\0—b)t-\-u is resorted to, however 

 large the number may be that is operated upon, the said operation is repeated until 

 only one digit remains, thus yieldmg the remainder to 8 without the performance 

 of any division. When 8=9 the operation consists merely of the addition of the 

 digits of the given number, reducing the result from time to time to a single figm*e 

 as may be requisite, also by addition ; for other values of 8 less than 0, multiplica- 

 tion as well as addition is necessary. The name imitation has been given to this 

 class of operations, the remainders being umtates, and the diAisor (8) the base. 



Operations upon remainders being analogous to operations (of the same kind) 

 upon dividends, an operation (unitation) in which the base has any value less than 

 10 (and certain values above 10) becomes available to verify arithmetical operations. 

 Also the unitate of an unkno'mi number may be calculated from a known number 

 with which it is connected by certain known operations. 



The following remarks will facilitate the practical use of the operations com- 

 prised imder the above-mentioned formulre, and will illustrate and suggest applica- 

 tions of the theorem that might otherwise remain dormant : — 



The general form of the notation to indicate the unitate of a number (.r) to the 

 base 8, is \] x=y, in which y is necessaril}"^ equal to or less than 8. As IJj.r is the 



simplest series of miitates that are useful, the suffix is left out, thus U.r. 



a:=28(25=-25) = 16G0-4, IT.r=Ul(4-5)=U(13-5) = 8; also U16G04 = 8. 



U,.r = U 7(U,4--U,2')=U,7(9-4)=7; also U,16604=7. 



Formulas involving direct operations in decimals may be checked in the same 

 way as other formulaj. 

 Example : 



.r=(2-8--54)(51-o-f--5=--7^)=11617982. 

 m-=U(10-9)(2+U5=-U73)=Ul(2+7-l)=8. 



Formula3 involving indirect operations, whether of decimals or otherwise, must 

 have the remainders of di\'ision or other terminations of the process taken fully into 

 account. 



Examjile : 



.T=fJ=2-00588A, U^=U?=1; 

 17 17 S ' 



