Sir Witt1Am Rowan Haminton’s Researches respecting Quaternions. 235 
the quotients thus denoted being numerical; we have, by article 16, for the case 
n = 2, the commas in the compound indices being here omitted for the sake of 
conciseness : 
Ag = Coo % Con B53 — My = Con Bo Con 15 I (186) 
Bry = Cro By Cy 2,3 yy = Cyy9 Fy Oy. 2,5 
Ay = Mp Ay FM, Ay 5 A = My Ay + M, 4,3 (187) 
i Loo Ogg + lo a= Lig Ay == Ln ays } (188) 
OH Mg Qi H bo @n = bho Gog Ay Gon 5 
and, consequently, 
ling = bog A A lo, 15 ] (189) 
Im, = ly a, + 1, a. J 
Again, by article 17, for the same case » = 2, we have the analogous formule : 
i , ing on = 1 le, 
qq = Cop Bq = Cop, 13 — By = Cor BH Con 15 (190) 
ic es. ‘ = ore ’ PS 
By = Cy FC M3 — Ar = Cy % A Cn U3, 
Ay = Mo Ay + IM; Bios A = May, + May; (191) 
and then, assuming these other expressions, 
ay = My Ay + mi a3 a = mM ay +m) a, (192) 
we find, by (188), two equations of the same forms as (189), namely, 
Img =, a0 +2, a1 5 } (193) 
Imi = 1,, a5’ +4, ay’. 
Making, therefore, according to the general rule contained in the formula (150), 
ip Snel Cues Gre 
= (Jpn Cr00 bby C10) eo Cora Cros tba Cyrn) Any (194) 
we have results included in the formula (149), namely, 
Mi; = S12, Mp Neos My ==,» M.MoN,. 5 (195) 
that is, more fully, 
Tk fares / p, / pat) : 
My, = My Mp Nog) My My NMyyg FM, My Rigg FM, M, Niyo 5 | (196) 
My’ ZS MyM Nyy + MyM, My, FM, IM; Nyy FM, MN, J 
