TRANSACTIONS OF THE SECTIONS. 527 



The value oi f~^ a, corresponding to a particular i in (4.), 

 he denotes by f7 ' a. There is a discontinuity in fj * a or 

 f~ (r + -v/ — 1 *), as above defined. When r is negative, 

 f7 {'' '^ \/ — \ s) is suddenly diminished by (quam proxime) 

 2 TT on the completion of the passage of s through from po- 

 sitive to negative. For the purposes to which the author ap- 

 plies f~ a, it is not necessary that for all nascent and trans- 

 itive, as well as finite and quantitative states and values of 

 the r and s and the r and s^ belonging respectively to a and 

 a, it should be predicable absolutely that f~ a, as above de- 

 fined, is the same individual function of a and i, thaty r a is 

 of a and the same i. It is sufficient for him, that, in all ima- 

 ginable cases, f~ a, when i is supposed to be arbitrary, com- 

 prises all the roots of the equationyfl = a, and, when i is sup- 

 posed to be individualized, denotes a unique value. These lat- 

 ter objects are attained by his notation, as above explained, 



which arbitrarily defines -—= to mean 1, whenever * = 0. 



That value of «■* which is expressed hyf(xf^ ' «\ he de- 

 notes by the symbol a* , and terms the i*'^ value of o'^ : o* is 

 an individual solution of ^^ in equation (1.) ; a^ and af are 

 similar individual exponential functions of x and x\ in a sy- 

 stem where a. is equal to a, and independent of i. The theo- 

 rems contained in the author's paper depend upon the original 

 definitions and principles assumed; and if different subsequent 

 definitions, subservient only to notation, were employed, — if 

 a value of «■" diflPerent from his a^ were arbitrarily assumed as 

 the primitive, the same theorems would still exist, though they 

 might require to be differently expressed. He gives sym- 

 metrical converging developments and easily calculable formulae 



for the real and imaginary parts of {r + ^/ — Is), ' 



X and .r' being real as well as r and *. 



Second. The second problem is to find a such that a" = y, 

 X and y being given quantities, real or imaginary. 



The general solution is 



«=/(— /"'y) = y"^. (5-) 



for y will certainly be found among the values of a', when a 



