183 



we can deduce from them four others, a', h', c', d', by the expressions, 



dialed') = f\])cd\ h'\_dbcd~\ = - f\cdcb\y &c., 



where /is used as the characteristic of a linear or distributive quaternion 

 function of a quaternion, of which the form is supposed to be given ; and 

 thus the general form of mcli a function conies to be represented by the 

 expression, 



. . r =fq = a^Saq + h'Shq + c'Scq + d'Sdq ; 



involving sixteen scalar constants, namely those contained in a'Vc'd'. 



The ProUem is to invert this function f\ and the solution of that 

 problem is easily found, with the help of the new Lemmas I. and II., 

 to be the following : — 



VL . . qi^alcd) {a'h'c'd') = {ahcd) {a'h'&d')f-W = \lcd~\ (^rVc'cV) 



+ \cda\ [red' a') + [tdaV] {rd'a'h') + [ahc'] {ra'h'c) ; 



of which solution the correctness can be verified, d posteriori, with the 

 help of the same Lemmas. 



Although the foregoing problem of Inversion had been virtually re- 

 solved by Sir W. R. H. many years ago, through a reduction of it to the 

 corresponding problem respecting vectors, yet he hopes that, as regards 

 the Calculus of Quaternions, the new solution will be considered to be 

 an important step. He is, however, in possession of a general method 

 for treating questions of this class, on which he may perhaps offer some 

 remarks at the next meeting of the Academy. 



The Secretary announced the following donations to the Museum : — 



1 . A medal struck in honour of Prederic Thiersch : presented by 

 the Royal Academy of Sciences of Eavaria. 



2. A commemorative medal: presented by the Royal Society cf 

 Christiania, I^orway. 



3. A stone ball and collar, found in a limestone gravel pit: pre- 

 sented by Hugh Blackney, Esq., Baliyellen, Goresbridge. The stone ball 

 weighs about six ounces, and measures six inches in circumference;, is 

 shghly oval, and fits the collar exactly. 



4. A small cannon-ball, weighing 2 lb. 14 oz., found on the battle- 

 field of Aughrim : presented by Dr. Bigger, 



5. A portion of a very flat stone " celt " foand in a turf bog at Con- 

 nemara : presented by Dr. Mac Swiney, Stephen' s-green. The celt is of 

 peculiar interest, as it retains on the weathered surfaces of its cutting 

 edge the scratches or marks of the fine sand with which it appears to 

 have been sharpened shortly before it was lost. 



6. A specimen of yellow tile, or brick, from the foundation of a 

 building at the corner of Grafton-street and JN'assau- street, described in 

 Mr. Mallet's note accompanying the donation. 



