SEPTEMBER 15, 1899.] 
previously been pointed out by Gregory as 
an independent principle, he called the 
law of association. As the principle of 
commutation was still assumed to apply to 
the terms of a sum, it followed that the 
principle of association also applied to 
them. Here, then, we have an important 
difference in the inventory of the laws of 
algebra. According to De Morgan algebra 
follows all the laws which he enumerated, 
and them only; but Hamilton showed that 
the legitimate extension of algebra to space 
requires the commutative law to be modi- 
fied in the case of a product. And still 
further, light is obtained on the nature of 
these laws by considering the way by 
which Hamilton satisfied himself of the 
truth of the principle of association. He 
sought for and obtained a geometrical proof, 
independent of the principle of distribution, 
and depending on theorems taken from 
spherical trigonometry or spherical conics. 
Thus a notable generalization of algebra 
was made, not by arbitrary choice of fun- 
damental rules, nor by arbitrary extension 
of the rules for integer number, but by find- 
ing out the universal properties of the sub- 
ject analyzed. 
We have already found that the index 
operations form a valuable test of the 
soundness of any theory of algebra. If 
the method of quaternions is the true ex- 
tension of algebra tospace we expect it to 
throw new light on these operations. Asa 
matter of fact, most of the works on quater- 
nions ignore the subject or present instead 
the treatment for the plane. In Hamil- 
ton’s ‘ Elements of Quaternions’ there is a 
chapter headed ‘On Powers and Logarithms 
of Diplanar Quaternions,’ but what it con- 
tains is practically limited to the plane. 
Why? Because the author believed, and 
there states, that the fundamental exponen- 
tial law is not true for diplanar quaternions ; 
that is, for space 
eP x ef not = eP*4, 
SCIENCE. 
305 
The source of error lies in regarding the 
sum of indices as commutative, for that 
amounts to holding that e? x ef =e’ x e*, 
which is contrary to the principles of 
quaternions. Were p+q a sum without 
any real order of the terms, then we might 
have an order of factors, that is, we might 
have 
(pP+q)(ptQD =P +p9t+npteg=pt 
q + 28pq. 
But when the sum has a real order of p, 
prior to q, then we cannotat the same time, 
hold that one factor p+q can be prior 
to another factor p+ q; for in the expan- 
sion we should have the contradiction of p 
being prior to q and q at the same time prior 
to p. Hence when p is prior to q the second 
power is not formed in accordance with 
the distributive principle; it is p? + 2 pq + q’. 
When this is admitted the exponential 
principle stands, but the commutative prin- 
ciple for a sum of such indices goes, as 
does also the distributive manner of form- 
ing the power of such a sum. 
As regards the third index law it is evi- 
dent from the non-commutability of the 
factors in general that in space it ceases to 
be true. The rule of reduction fora sum of 
terms requires to be modified when the 
terms have a real order; for p+q—q=p, 
but ¢ +p—qis not equal top. The term 
and its opposite must follow one another 
immediately in order that the reduction 
may be legitimate. Similarly, in the case 
of a product the factor and its reciprocal 
must follow one another immediately in 
order that the reduction may be legitimate. 
From these principles the generalization for 
space of all the fundamental theorems of 
algebra follows without difficulty, and the 
theory of logarithms and exponents becomes 
the most fruitful part of quaternion analysis. 
We may now consider briefly how the 
advance made by Hamilton struck a co- 
temporary mathematician—Professor Kel- 
