202 THIRD REPORT 1833. 



expressions as m a in symbolical algebra, when w/ is a number 

 whole or fractional, and a any symbol whatsoever. When m, n 

 and a are whole numbers, it very readily appears that ma -\- na 

 = {m + n) a, and that ma — n a = {m — n) a : the same con- 

 clusions are true likewise for all values of ?n, n and a. In 

 arithmetical algebra we assume a^, a^, a'*, &c., to represent a a, 

 a a a, aaaa, &c., and we readily arrive at the conclusion that 

 a" X a" = a'"'^ ", when w and 7^ are whole numbers : the same 

 conclusion must be true also when m and n are any quantities 

 whatsoever. In a similar manner we pass from the result 

 (a*") " = «""•, when « is a whole number, to the same conclusion 

 for all values of the symbols *. 



The preceding conclusions are extremely simple and element- 

 ary, but they are not obtainable for all values of the symbols 

 by the aid of any other principle than that of the permanence 

 of equivalent forms : they are assumptions which are made in 

 conformity with that principle, or rather for the piirjiose of 

 rendering that principle universal ; and it will of course follow 

 that all interpretations of those expressions where m and n are 

 not whole numbers must be subordinate to such assumptions. 



Thus, 7T + "o' = (~rt+';T) a •=■ a, and therefore -^ must 



mean one half of a, whatever a may be ; ax a = a* ' ' 



= a = a, and therefore a must mean the square root of a, 

 whatever a may be, whenever such an operation admits of 



interpretation. In a similar manner -3- must mean one third 



part, and a' the cube root of a, whatever a may be, and simi- 

 larly in other cases : it follows, therefore, that the interpreta- 

 tion of the meaning of a , a^, &c., is determined by the general 



* The general theorems ma -\- na=z{m-\- n) a and ma — « a = {m — n) a, 



a"* X 0"= 0"* + " and — = a*"-", (a*")" = a'"" and (a"") » = a " > which 



are deduced by the principle of the permanence of equivalent forms, and which 

 are supplementary to the fundamental rules of algebra, are of the most essen- 

 tial importance in the simplification and abridgement of the results of those 

 operations, though not necessary for the formation of the equivalent results 

 themselves. It also appears from the four last of the above-mentioned theorems 

 that the operations of multiplication and division, involution and evolution, are 

 performed by the addition and subtraction, multiplication and division, of the 

 indices, when adapted to the same symbol or base. If such indices or logarithms 

 be calculated and registered with reference to a scale of their corresponding 

 numbers, they will enable us to reduce the order of aiithmetical operations by 

 two unities, if their orders be regulated by the following scale ; addition (1), sub- 

 traction (2), multiplication (3), division (4), involution (5), and evolution (6). 



