21] 
127 
21. 
i) 
ON JACOBI’S ELLIPTIC FUNCTIONS, IN REPLY TO THE 
REV. B. BRONWIN; AND ON QUATERNIONS. 
[From the Philosophical Magazine, vol. xxvi. (1845), pp. 208, 211.] 
The first part of this Paper is omitted, see [17]: only the Postscript on Quaternions, pp. 210, 211, is printed. 
It is possible to form an analogous theory with seven imaginary roots of ( 
(? with i/ = 2 M —1 roots when v is a prime number). Thus if these be 
which group together according to the types 
123, 145, 624, 653, 725, 734, 176, 
i.e. the type 123 denotes the system of equations 
i\i‘> = is > i$3 = i\ > hh = i‘l > 
ifll — is, i$2 = i\ > i\i 3 — i‘2> 
&c. We have the following expression for the product of two factors: 
(X 0 4- X 1 i x + ... X 7 i 7 ) (X' 0 + X\ i x + ... X' 7 i 7 ) 
il> i‘2} i 3 > i\i if» is 
= 
AoX'o 
- 
M 
- 
X 2 X' 2 
... - 
X 7 X\ 
+ 
[23 
+ 
45 
+ 
76 
+ 
(01)] t. 
+ 
[31 
+ 
46 
+ 
57 
+ 
(02)] in 
+ 
[12 
+ 
65 
+ 
47 
+ 
(03)] ¿3 
+ 
[51 
+ 
62 
+ 
47 
+ 
(04)] f 4 
+ 
[14 
+ 
36 
+ 
72 
+ 
(05)] i B 
+ 
[24 
+ 
53 
+ 
17 
+ 
(06)] i 6 
+ 
[25 
+ 
34 
+ 
6l 
+ 
(07)] ¿ 7 
(01) = 
XoX’i 
+ X x X' 0 ... 
1 > 
12 = X 1 X , 2 
- X.x; 
where 
and the modulus of this expression is the product of the moduli of the factors. The 
above system of types requires some care in writing down, and not only with respect 
to the combinations of the letters, but also to their order; it would be vitiated, e.g. by 
writing 716 instead of 176. A theorem analogous to that which I gave before, for 
quaternions, is the following:—If A = 1 +X 1 i 1 ... X = x x i x ... + x 7 i 7 : it is immediately 
shown that the possible part of A -1 A A vanishes, and that the coefficients of i x ,... i 7 
are linear functions of a^, ... x 7 . The modulus of the above expression is evidently the 
modulus of X; hence “we may determine seven linear functions of x x ...x 7 , the sum 
of whose squares is equal to xf + ... + # 7 2 .” The number of arbitrary quantities is 
however only seven, instead of twenty-one, as it should be.