Sir Witt1am Rowan Hamirton’s Researches respecting Quaternions. 271 
be satisfied, where c is a positive or negative number, then the equation (371) 
will be satisfied by either of the two quaternions which are included in the 
following expression, and by no other quaternion, 
q= (wu, 0,70; 0) = (a= (a —5); 0; 00). (376) 
The same expression holds good, giving one solution of the equation (371), for 
the case a = 6. But in the remaining case, where 
Ci<e0 a 06a (oii) 
c being still a positive or negative number, we are to adopt the remaining alterna- 
tive (373), namely, w =a; and instead of supposing r = 0, we are now, by the 
first equation (372), and by (377), to suppose 
7S = Pp oe) 0 Se (378) 
and the solution of the quadratic equation (371) is now expressed by the par- 
tially indeterminate quaternion, connected with the two couple-solutions (a,c), 
q = (a, 2, y, 2), where 2? + 7?-++ 2? = 5b — a’. (379) 
And thus we may perceive that, if we denote by » the modulus of the first 
numeral quaternion (299), which may represent any such quaternion, then this 
quaternion, q, is a root of a quadratic equation, with real coefficients, namely, 
the following : 
‘ q —2uq+p=0. (380) 
Exponential and Imponential of a numeral Set; general Expression for a 
Power, when both the Base and the Exponent are such Sets. 
39. The investigations, in some recent articles, respecting certain powers and 
roots of a quaternion, may be made at once more simple and more general by the 
‘ introduction of a well-known exponential series. We shall, therefore, write 
n a M3 i 
pgy=14+t4+4 ro3 t ke (381) 
and shall call this series the exponential function, or simply, the exponential of 
the numeral set q, with respect to which the operations are performed ; we shall 
also denote this exponential still more concisely by writing simply pg instead of 
P(g), where no confusion seems likely to arise from this abbreviation. The 
inverse function, which may be conceived to express reciprocally g, by means of 
VOL. XXI. 20 
