22 



Proceedings of the Royal Irish Academy. 



then oca is the price of one egg, and ya the quantity I get for a shilling, the 



problem states 



yxd = 12a', 



</+2)(.v-^)]«'=12«'; 



12 



y = — ; 



"S-')K 



12 o' = 0, 



6 + . 



12/ - 6 i a ' - °> 



- + 



6.B- 



1 

 iV 



1 



~ 2 



0.H a' 



= 



(■ 



24, 



)'- 



V24J 



1) , 

 - 2 j- 



= 



1 



17 



\f 



1 



17\ , 





24 



24 



)(' 



~24 + 



2lJ a 



= 



.8 

 !4 = 



o 

 O 



= 4' 



or 



., = - 



16 

 24" 



2 

 3' 



(Observe, we can proceed backwards from either « = f, or a; = -f to the 

 two original equations.) Thus the price I pay for 12 eggs is 12.ro' = 9a' or 8/3' ; 

 so either I pay 9 pence or am paid 8 pence to take the eggs. 



Indices. — The theory of indices may now be developed in the usual way. 



"We define -/'• when m and n are positive integers, and — is in its lowest 



m J 



terms as "J.e"', and * » as—- "When x is positive, "JjP has two values 



a'"» 



equal but of opposite sign when n is even, and one if n is odd. Also, we 



in m' 



prove generally tbat a value of .<■" multiplied by a value of x"' gives a value 



mil' + nt'n 

 Of X mn 



If x is negative, "Jx"' does not exist as a real number when on 

 is odd and n even ; it lias one value when m is odd and n odd, and one value 

 when m is even and n odd. "We note, then, the theory is incomplete so long as 

 we restrict our operations to real numbers. The binomial exponential and 

 logarithmic series may now be developed, noting that we restrict ourselves to 

 the logarithms of positive numbers. 



Trigonometry. — In trigonometry, angles are measured in terms of a unit a, 

 the radian, in the form xo, where # is a real number. Associated, then, with 

 any real number x are other real numbers sin xa, cos xa, &c. 



