276 



is found to be satisfied, we may then infer that the five cor- 

 ners A, B, c, D, E, of this pentagon, are situated on the surface 

 of one common sphere. This equation of homosphcericism, 

 (5) or (6), appears to the present author to be very fertile 

 in its consequences. To leave no doubt respecting its meaning, 

 and to present it under a form under which it may be easily 

 understood by those who have not yet made themselves mas- 

 ters of the whole of the theory, it may be stated thus : if we 

 write for abridgment, 



a, = i {Xy - *'2) +j (y, — Vi) + ^ (z, - Zi), 



02 = ^■ {x^ - x^) + j («/2 - t/s) + ^ (2^2 — 2^3). 



03 = i {x^ — x^ +j {1/3 — 2/4) + A (S3 — Zi), 

 a^ = i (Xi — x^) +7 (2/4 — ys) + A (z^ - Zg), 

 a5 = i {x^ — a;,) + J («/5 — 2/1) + A (z^ — z,). 



(') 



and then develope the continued product of these five expres- 

 sions, using the distributive, but not (so far as relates to ij k) 

 the commutative property of multiplication, and reducing the 

 result to the form of a quaternion, 



ax aa 03 04 05 = w + ia: +jy + kz, 



(8) 



by the fundamental symbolical relations between the three 

 coordinate characteristics ijk, which were communicated to 

 the Academy by Sir William Hamilton in November, 1843, 

 and which may be thus concisely stated : 



i-" =:f=zk^ = ijk= - I; (A)* 



and if we find, as the result of this calculation, that the term 



* These fundamental equations between the author's symbols i,j, k, appeared, 

 under a slightly more developed form, in the number of the London, Edinburgh, 

 and Dublin Philosophical Magazine for July, 1844; in which Magazine the 

 author has continued to publish, fi-om time to time, some articles of a Paper on 

 Quaternions; reserving, however, for the Transactions of the Royal Irish 

 Academy, a more complete and systematic account of his researches on this 

 extensive subject. 



