TRANSACTIONS OF SECTION A. 529 



Hence, in either case, if one set (P^ P^,, &e., P,) be given, and also /,/', the 

 other set becomes known. 



Also a simple relation exists between P„,j, P,,^- where P,, = i2-''^\ Mj, . F, 

 P,,'-=2=«'-'.M,,/.F'; then will m = / (if .7 = ^' + 2), and r = L,. .F', provided 

 q,M,„Lr/. y', M,;,, L,,. be all prime and prime to F, F', where M,j = 2''-1 and 

 L, = ^(,2" + l). 



Hence if P„,' be given, P^, is also known — Ex. {q', q) = (5, 7) or (17, 10) 



9. On a Practical Rule for finding the Perimeter of an Ellipse. 

 By Thomas Mum, G.M.G., F.R.S. 



10. The late J. Hamhlin Smith's Rule for the Decimalisation of English 

 Money} By J. D. Hamilton Dickson. 



To divide by 96 is equivalent to multiplyina: by the aeries - - + ^ + + . . . 



The late Mr. .7. Hamblin Smith discovered a remarkable application of this to 

 converting shillings and pence into decimals of a £. Starting with the two known 

 first steps, namely, ' multiply the shillings by 5 and write the result in the first 

 two places after the decimal point, then multiply the pence and \ pence by 4 

 and add the result (increased by 1 for 6c?. or over) to the second and third places 

 after the decimal point,' he proceeds thus: ' Multiply the last two decimals found 

 at any stage of the process, after the first three have been obtained as above, 

 by 4, take the digit in the ten's place of the product, append it to the decimals 

 already found as the next decimal, and repeat the process ; with the proviso that 

 should the 4-product end with 48, 68, or 88, the digit in the ten's place is to be 

 taken as 5, 7, or 9 respectively.' For example. 2Z, 7s. 1\d. gives at successive 

 steps 2'382, (4x82 = 328 hence) 2-3822, (4x22 = 88 hence by proviso) 2-38229, 

 (4x29=116 hence) 2-382291, then 2-3822916, and now 6 repeats, so that the 

 result is 2-3822910/. 



A simple extension of the rule meets the case where (odd) eighths of a penny 

 come into the sum of money. Calculate the first four places of the decimal for 

 the sum which is one-eighth of a penny less than the given sum, now add 5 to 

 the fourth decimal place (carrying unit to the third decimal place if necessary), 

 and proceed according to the ordinary rule. 



The converse of these operations is obvious, only three (or in the last case 

 four) decimal places being necessary to get the answer correct to farthings. 



For any divisor near and less than 100 (say from 99 to 93) a similar process 

 is applicable, with a suitable change of the multiplier 4. Further, the process is 

 applicable to any dividend by a slight extension of the list of provisos. The rule 

 is, to divide any number by one of these divisors (say 97), multiply the last two 

 digits of the quotient found at any stage of the process by the associated multiplier 

 (here 3), add the digits in the ten's and unit's places of the product to the next 

 two following digits of the dividend, considered as being in ten's and unit's places 

 as they stand, take the digit in the ten's place of this sum and append it to the 

 part of the quotient already found, thus extending the quotient by one digit, then 

 repeat the process ; subject to the proviso that if the last two digits in the sum 



97 



68, 78, 88, 98 



39, 49, 59, 69, 79, 89, 99 



be contained in the annexed table (for 97), the new digit to be added in the 

 quotient is to be 4, T), 6, 7, 8, 9, t respectively, where t is 10. There are similar 

 tables for other divisors. 



' Transactions Edinhurgli Matlwmatioal Society, 1902-1903. 

 1902. M M 



