TRANSACTIONS OF SECTION A. 503 



showed that this form of the differential equations possesses great importance for 

 Analytical Mechanics. The marked attention, therefore, of mathematicians could 

 not fail to be aroused when Herr Hamilton, Professor of Astronomy in Dublin, 

 indicated in the " Philosophical Transactions " that in the Mechanical problem 

 referred to all the integriil equations of motion mioht be represented in just as 

 simple a manner by means of the Partial Differential CoelHcienta of a single 

 function. This is undoubtedly the most considerable extension which Analytical 

 Mechanics has received since Lagrange.' 



It will be of interest to the t-ection to recall the fact that Hamilton and Jacobi 

 met each other for the first and I fancy the only time at a Meeting of this 

 Association, held in Manchester in 1842, at wliich meeting Jacobi, addressing this 

 Section, called Hamilton ' le Lagrange de votre pays.' 



The last third of Hamilton's life was mainly devoted to the development of his 

 Quarternion Calculus, As early as 1828 his Class Fellow, J. T. Graves, who had 

 been working at the theory of the use of imaginary quantities in Mathematics, 

 wrote an essay on Imaginary' Logarithms which he wished to get printed by the 

 Royal Society. There appears to have been some hesitation amongst the leading 

 mathematicians in the Society, notably, Herschel and Peacock, about publishing 

 Graves' paper, as they felt dubious about the accuracy of his reasoning. Hamilton 

 heard of this and wrote earnestly to Herschel defending his friend's conclusions, 

 and it seems as if his generous desire to help his friend tirst set his own mind 

 working in this direction. 



For years his busy brain in the midst of all his other work kept pondering over 

 this question of the interpretation of the imaginary, and he has left us in his 

 ' Lectures on Quaternions ' an elaborate account of the many s^'stems he devised. 



It was only in 1843, fifteen years later, that he first invented the celebrated 

 laws of combination of the quadrantal versors of the Quaternion Calculus, 

 Argand, Cauchy, and others had proposed for space of two dimensions the theory 

 now known as that of the Complex Variable, For them ,r + iy meant the vector 

 to the point .ly, and the product of two vectors meant a new vector of the same 

 form, the only law required being that i operating upon i was always equivalent 

 to -1. 



Many attempts had been made to form on similar lines a Calculus which 

 should apply to space of three dimensions ; but so far all such attempts had proved 

 unsuccessful, the laws by which the new symbols acted upon one another leading 

 to results hopelessly involved. It was here that Hamilton's wonderful faculty of 

 scientific imagination came into play. He proposed that a vector should be 

 denoted by ix+jy + kz. As in the theory of the complex variable in two dimen- 

 sions the result of any number of successive operations always preserved the 

 fundamental type a + ib, so it was desirable that the result of the successive 

 operations of his vectors should issue in an equally simple fundamental type. This 

 end he found he could attain if he discarded the commutative principle which 

 hitherto had barred his own progress and that of others, yet preserving the 

 distributive and associative principles, and finally one happy evening he arrived 

 at the beautifully simple laws by which the symbols of this Calculus act upon 

 each other ; that not only i" =j- = ^'- = - I, but also that rj = —ji = k,jk = — ^7 = i, 

 ki— —ik=j. 



Though it was thus — as the product, that is, of two vectors — that the Quaternion 

 first presented itself to Hamilton, he of course saw that it immediately followe4 

 that it might be regarded as the ratio of two vectors, in other words the operation 

 which turned one vector into another. In fact in the more synthetic expositioi^ 

 which is contained in ' The Elements ' he makes this latter the starting definitior^ 

 of the Quaternion. 



It iy noteworthy that this the more complete and systematic presentation of 

 the subject b}'^ its illustrious autlpr may be said to owe its origip to the keen 

 interest my predecessor, Professor Tait, took in the new Calculus, of which, as you 

 know, he ever afterwards remained the most ardent champion. This interest le(| 

 him to seek from Dr. Andrews an introduction to Hamilton, and the encourage; 

 pient came to Jlamjlton at ap opportune paoment, for be vyypte ;-r- 



