310 Mr. Cayley oti Quaiemions. 



Mr. Bronwin wishes me to do, by expressing 3/ by means of 

 the complementary functions, and find that the process agrees 

 step by step with Jacobi's, the only difference being that the 

 transformation to the complementary functions is there made 

 at the end. 



An extraordinary assertion is the following one: — "It is 

 sufficient to observe that the first form of w only will satisfy 

 the conditions sa.u = 0, sa.zi=l, required by Jacobi's 

 theoi'y, pages 40 and 41." If w is a misprint for v, and these 

 equations are to be satisfied for u = 0, Jacobi's form of sa.v 

 certainly satisfies them in any case; the first, because of 

 the factor s .a .u; the second, because M is just determined 

 by this very condition. If this is not the meaning, the true 

 one has escaped me. One word on my preceding paper : the 

 principal thing gained in it seems to be, its being likely to 

 lead to the complete determination of the values of A, A' in 

 the general case, a question which Jacobi has not examined ; 

 the principle is very clear, and one that is immediately sug- 

 gested by Abel's formulae, but I have no wish to force it upon 

 Mr. Bronwin. I remain, Gentlemen, 



Your obedient Servant, 



Cambridge, January 16, 1845. A. Cayley. 



P.S. On Quaternions. 



It is possible to form an analogous theory with seven ima- 

 ginary roots of ( — 1) (? with v = 2"— 1 roots when v is a prime 

 number). Thus if these be <i, «2, jg, »4, »5, ig, j^, which group 

 together according to tlie types 



123, 145, 624, 653, 725, 734, 176, 

 i. e. the type 123 denotes the system of equations 



*i '2 — *3» '2 '3 — *i» '3 '1 ~ '2' 



'2'l~"~'3» '3'2— ~~'l' *1 '3~ '2> 



&c. We have the following expression for the product of two 

 factors : 



= XoX'o— 2^, X'j — X2X'2 ...— XyX'y. 



+ [23+45 + 76 + (01)]«i where (01) = XoX', + XiX'q 



+ [3T + 46 + 57 + (02)]i2 :_ 



+ [12 + 65 +47 + (03)] »3 12 rrrXiX'a-XgX', 



+ [5T + 62 + 47 + (04)]«4 &c. 



+ [r4 + 36 + 7"2+(05)]<5 



+ [24 + 53+17 + (06)]*(; 



+ [25 + 34 + 61+ (07)] «7 



