182 



The Secretary, on the part of the Rev. CharlesYignoles, Yiear of 

 Clonmacnoise, presented rubbings of three ornamented stones lately dis- 

 covered at Clonmacnoise, one of which bears the inscription Op com 

 sari. 



The thanks of the Academy were voted to the donor. 



MONDAY, JUNE 9, 1862. 



The Yeey Eev. Charles Graves, D. D., President, in the Chair. 



The Rev. Dr. Reeves read a paper concerning the Identification of 

 St. Molagga's Church of Lann Beachaire, in Fingall, with the Ecclesi- 

 astical Remains at Bremore, in the parish of Balrothery, a little north 

 of Balbriggan, which bear the name of Lambeecher in the Liber ^[iger 

 of the See of Dublin." 



Sir William R. Hamilton, LL. D., read the following paper : — 



On a JN'ew and General Method of Inverting a Linear and Qua- 

 ternion Etjnction of a Quaternion. 



Let a, h, c, d, e represent any five quaternions, and let the foUovv^ing 

 notations be admitted, at least as temporary ones :— 



ah - ha = ; 8[_ah']c - (ahe) ; 



{ahc) + Ich'jSa + [^ac'jSh + [_ha]8c = ; 



Sa[hcd'] = (ahcd) ; 

 then it is easily seen that 



[«^&] = - \l)a] ; (ahc) = - (hac) - {hca?) = &c. ; 



[^ahc'] = - \^hac'] = [hca'] = &c. ; 



(ahcd) = - (hacd) = (head) = &c. ; 



0 = [^aa^ = (aac) = [aac'l = (aacd), &c. 



We haA^e then these two Lemmas respecting Quaternions, which 

 answer to two of the most continually occurring transformations of 

 vector expressions : — 



I. . . 0 = a(lcde) + h{cdea) + c(deah) + dieahc) + e(ahcd), 

 or V. . . e(ahcd) = a{ehGd) + h{aecd) + c(ahed) + d{ahce) ; 

 and II. . . e{ahcd) = [hcd^Sae - \^cda']She+ [^dah']Soe — \_ahGlSde; 

 as may be proved in various ways. 



Assuming therefore any four quaternions h, c, d, which are not con- 

 nected by the relation, 



(ahcd) = 0, 



