14«4< Mr. Cayley on certain results relating to Quaternions. 



Sec. for determinants of any order, whence the theorem, if any 

 four (or more) adjacent vertical cohmins of a quaternion deter- 

 minant be transposed in every possible manner, the sum of all 

 these determinants vanishes, which is a much less simple pro- 

 perty than the one which exists for the horizontal rows, viz. 

 the same that in ordinary determinants exists for the horizontal 

 or vertical rows indiiferently. It is important to remark that 

 the equations 





"UJ 



■■0 or 



= 0, &c. . . . (13.) 



ra", vj' 



ue. rs <^' —w' <^ = 0, or ro-^' — <f)Z3-' = 0, &c. 



are none of them the result of the elimination of II, <I>, from 



the two equations 



On the contrary, the result of this elimination is the very dif- 

 ferent equation 



rar-l.(f>-OT'-1.4i'=0, (15.) 



equivalent of course to four independent equations, one of 

 which may evidently be replaced by 



Mot. M <fi'— Mot'. M 45 = 0, . . . (16.) 



if M GT, &c. denotes the modulus of m, &c. An equation ana- 

 logous to this last will undoubtedly hold for any number of 

 equations, but it is difficult to say what is the equation analo- 

 gous to the one immediately preceding this, in the case of a 

 greater number of equations, or rather, it is difficult to give 

 the result in a symmetrical form independent of extraneous 

 factors. 



I may just, in conclusion, mention what appears to me a 

 possible application of Sir William Hamilton's interesting dis- 

 covery. In the same way that the circular functions depend 

 on infinite products, such as 



-n(i + ^),&c., . . . . . (17.) 



\_m any integer from 00 to — 00 , omitting m = 0] 

 and the inverse elliptic functions on the doubly infinite pro- 

 ducts 



^nflH- ^ :), &c (18.) 



[w and n integers from co to — co , omitting w = 0, « = 0], 

 may not the inverse ultia-elliptic functions of the next order 

 of complexity depend on the quadruply infinite products 



