Nov. 13, 1884] 



XATURE 



enormous mechanical movements under which, as the 

 rocks sheared, the individual particles were forced over 

 cat b other in one common direction, viz. from east- 

 south-east to west-north-west. Further evidence of this 

 mechanical movement is supplied by certain abundant fine 

 parallel lines, like those of slickensides, which occur almost 

 everywhere on the foliation-surfaces or other parallel 

 divisional planes. They are especially well developed 

 among the striped fissile schists and the gneissose flag- 

 stones. These lines run in the same general direction 

 already mentioned (E.S.E. to W.N.W.). In many 

 cases it may be observed that the component particles 

 of the rocks are oriented in this same direction, while 

 original quartz-veins are drawn out into parallel rods. 

 Another important feature connected with these rocks 

 is the development of minerals along the new planes of 

 schistosity. In particular, the abundance of seriate mica 

 is noteworthy, the longer axes of the crystals of which lie 

 in a direction parallel with the slickenside-lines. Other 

 micas, hornblende, actinolite, and garnets have also 

 made their appearance along the same planes. This re- 

 crystallisation becomes more pronounced the further ca^t 

 one advances from the outcrop, or passes upwards from 

 the great thrust-plane. 



This accumulated evidence points to the conclusion 

 that in the north-west of Sutherland the rocks have been 

 powerfully affected by one grand series of terrestrial 

 movements whereby new structures have been super- 

 induced upon them. Among these changes the original 

 characters of the rocks have been more or less completely 

 effaced, and new crystalline structures have been produced. 

 Although a normal upward succession from the Silurian 

 strata into an overlying series of schists cannot be main- 

 tained in the north of Sutherland, it is nevertheless certain 

 that the displacements and metamorphism here described 

 are later than Lower Silurian time. It is also evident that 

 these great changes had been completed before the time 

 of the Lower Old Red Sandstone, the conglomerates and 

 breccias of which rest upon and are made up of fragments 

 of the crystalline schists. 



One final feature of the Durness and Eriboll area rem lins 

 to be noticed. The geological structure of this region has 

 been further complicated by the subsequent folding of the 

 strata, and by a double system of normal faults (// in 

 Section) which affect the strata and thrust-planes alike. 

 One set of normal faults trends north-north-east and south- 

 south-west, while another, which appears to be newer, 

 trends more or less at right angles to these. By these 

 two systems of later dislocations, the thrust-planes with 

 their low hade have been intersected and shifted precisely 

 as if they had been ordinary boundary-planes between 

 '< igical formations. Much of the difficulty, indeed, 

 which has been from the first experienced in unravelling 

 the complicated structure of this region may be traced to 

 the effect of the intricate network of reversed and normal 

 faulting. The very preservation of the Durness Basin 

 is due to two normal step-faults, one of which lets down 

 the quartzites more than 1 200 feet, while the other brings 

 the whole Silurian Basin down to the sea-level. 



B. N. Peach 

 John Horne 



THE GEXESIS OF AX IDEA, Oh STORY OF A 

 DISCOVERY RELATING TO EQUATIONS IN 

 MULTIPLE QUANTITY 



T VENTURE, even at the risk of appearing egoistical, 

 ■*• to call the attention of a wider circle of English 

 mathematical readers than are likely to notice it in the 

 pages of the Comptes Rendu!, to what appears to me a 

 remarkable discovery in the theory of matrices, or, in 

 other words, of multiple quantity which has lately pre- 

 sented itself to me. It seems to me the more necessary 

 to do so because the nature of the theorem which 



constitutes the discovery would hardly be suspected 

 from the leading title of the note in the Comptes Rendus 

 in which it is contained, being indeed an after-thought, 

 so that the sting of the paper has to be sought for in 

 its tail. 



Hamilton, of immortal memory, has given, in his "Lec- 

 tures on Quaternions," a solution of a certain quadratic 

 equation in quaternions, those algebraical entities which 

 (building upon a suggestion in Prof. Cayley's ever-memor- 

 able paper 1 on matrices, in the Philosophical Transactions 

 for 1858 or thereabouts) I have, with the general concur- 

 rence of all who have given attention to the subject, found 

 me. ins of identifying with binary matrices or algebraical 

 quantities of the second order, and thus succeeded in 

 determining the True Place of Quaternions in Nature. 

 Now, what Hamilton has done for an equation of the 

 second degree of quantities of the second order, the 

 theorem in question effects in a much more simple and 

 complete manner for a similar sort of equation of any 

 degree and relating to quantities of any order. 



The history of the discovery in question constitutes 

 in itself, it seems to me, an interesting chapter in 

 Heuristic. This is how it came about. Hamilton's 

 equation admits of six solutions or roots, which arrange 

 themselves naturally in three pairs, and stand in im- 

 mediate, and what we algebraists call rational relation 

 to the three roots of a cubic equation, or rather to the 

 six square roots of those three roots. From this it fol- 

 lows immediately that one single condition must be suf- 

 ficient to reduce the number of distinct roots of the equa- 

 tion in quaternions or binary matrices from si* to four, 

 inasmuch as, when two roots of the cubic referred to 

 become equal, two pairs of roots of the original equation 

 must coincide. It naturally therefore became an object of 

 interest to obtain the quantity which, equated to zero, ex- 

 presses the condition of equality of two roots of this 

 cubic, which of course may be effected by means of a 

 well-known formula for finding the discriminant of a 

 cubic equation ; but the quantity so obtained directly 

 from the cubic is of an exceedingly complex form, and 

 leaves the mind entirely unsatisfied as to its true internal 

 composition, just as from a handful of diamond dust it 

 would be impossible to infer the crystalline form which 

 constitutes the true idea, the raison or/agon d'etre of the 

 glittering gem. 



Again and again my mind reverted fruitlessly to the 

 subject until, on September 28 last, pacing the deck of the 

 splendid Dover and Calais boat, the Invicta, under the 

 vivifying and genial rays of a bright and benignant sun, 

 the idea occurred to me that the form to be determined 

 must be subject to satisfy a certain partial differential 

 equation, and without the aid of pen or pencil I suc- 

 ceeded, ere the voyage was half over, in identifying the 

 discriminant of the cubic with that of a biquadratic of 

 the simplest imaginable constitution possible : in technical 

 language, supposing/ .r- + q x -\- >' = ° to be the equa- 

 tion in question, I discovered virtually that the desired 

 discriminant is identical with that of the biquadratic form 

 which is the determinant of the binary matrix (or the 

 tensor squared of the quaternion) p x 1 -j- qx -f- / treated 

 as if x were an ordinary quantity. Starting from this 

 point it was easy to infer all the possible cases of equality 

 which could occur between the six roots ; and, more than 

 that, to classify under thirteen classes all the principal cases 

 that could present themselves in the solution of the equa- 

 tion, not merely for the general case when there are six 

 actual and determinate roots, but even for those cases 

 when some of the roots pass off into infinity and become 

 conceptual instead of actual, or else, without passing to 

 infinity, remain actual but contain arbitrary constants. 



This more-than-anticipated complete solution of the 

 problem before me was in part suggested by the opening 



1 This paper constitutes a second birth of Algebra, its aratar in a n< 

 glorified form. See introduction t 



xth volume of the A 



Lectutes on Universal Algebn 

 Mathematical Journal. 



