SEPTEMBER II, 1902] 
NATURE 
479 
He knew Wordsworth intimately, and the poet, to whom 
he sent some of his productions, gives him the following candid 
advice :— 
**Tt would be insincere not to say that something of a style 
more terse and a harmony more accurately balanced must be 
acquired before the bodily form of your verses will be quite worthy 
of their living souls. You are perfectly aware of this, though 
perhaps not in an equal degree with myself; nor is it desirable 
you should be, for it might tempt you to labour which would 
divert you from subjects of infinitely greater importance.” 
Hamilton was first in his,College classes in every subject and 
at every examination, and it was fully expected that he would 
carry off both the medals in Mathematics and Classics at his 
Degree when the following circumstances suddenly changed all 
his plans. Dr. Brinkley, the Professor of Astronomy in the 
University, was appointed to a Bishopric, and Hamilton, though 
still an undergraduate, was invited to offer himself for the 
vacant Chair. Sir George Airy and more than one of the 
Fellows of Trinity were also candidates, but Hamilton was 
unanimously elected. 
His career as an original author dates from this time, for 
immediately after his appointment he communicated to the 
Royal Irish Academy the first of three remarkable papers on 
** Systems of Rays.” 
Two striking features may be observed in these papers, as 
indeed in all his scientific memoirs : the generality and compre- 
hensiveness with which he states his object at the outset 
and the confidence with which he follows the bold and original 
lines of treatment which he lays down for himself, and closely 
connected with this, the determination not to be bafiled 
by any laboriousness of calculations which the application 
of his method may involve him in. In his first paper he 
begins by examining what happens to a system of rays of 
light emanating from a point and subjected to any number of re- 
flections at curved surfaces. He establishes the theorem that 
such a system will be cut orthogonally bya system of surfaces, the 
length of the path measured from the original source to any of 
these surfaces being the same for all the rays. The proof he 
gives of this theorem is so simple that it now seems almost 
axiomatic ; but it is curious that Malus, who had made the laws 
of Light his special study, though he suspected that the theorem 
ought to hold, yet found himself unable to establish it. 
Hamilton, now considering the length of the path tc any 
point as a function of the coordinates of that point, and denoting 
this function by V, proves that V satisfies a simple partial 
differential equation of the first order and proceeds to show the 
important part the function V plays in the theory. 
He goes on to prove generally that if we are dealing, 
not with right lines, that is, with paths, for which as be- 
tween any two points fa is a minimum, but with curved paths 
for which | was is a minimum (where w is a function of the 
coordinates), and a system of such paths be drawn through a 
given point, O, the system of surfaces V=const. will still cut all 
the paths at right angles. If we adopt the emission theory of 
Light, and we take for u the velocity of Light, V becomes 
“the Action,” and the minimum property which the paths 
satisfy is the principle of ‘* Least Action.” If, on the other 
hand, we adopt the undulatory theory, and we take for u the 
reciprocal of the velocity, the minimum property becomes the 
principle of ‘‘ Least Time.” Thus Hamilton shows that, by 
altering the significance of «, his method applies to either 
theory. 
Introducing the further conception that u depends, not only 
on the coordinates of the point, but also on the direction-angles 
of the ray, he is able to apply his reasoning to rays passing 
through a crystal. He gives by his method a new and interest- 
ing proof of the equation of Fresnel’s wave-surface, and arrives 
at the conclusion, hitherto unnoticed by mathematicians, that 
this wave-surface possesses four conical cusps and also four 
special tangent planes, each of which touches the surface, not 
in one point only, but in an infinite system of points lying in a 
circle. The physical significance of these theorems is what is 
known as Conical Refraction. 
Having drawn this inference from his mathematical analysis, 
Hamilton wrote to his friend Dr. Lloyd and asked him to verify 
it by actual observation, and accordingly Hamilton’s paper in 
the Transactions of the Academy is accompanied by another 
NO. 1715, VOL. 66] 
‘of scientific imagination came into play. 
from Lloyd, describing the beautiful arrangements by which 
he had succeeded in verifying this remarkable phenomenon in 
both its varieties. 
This striking instance of scientific prediction naturally made 
a great sensation at the time, appealing, as it did, toa much 
larger public than the few select mathematicians who were 
capable of mastering the elaborate treatise on ‘‘ Systems of 
Rays. 
‘The experimental skill that was required to obtain these 
results may be realised from the circumstance that as I have 
been told the French physicists found themselves unable to 
repeat the experiment till Lloyd himself went over to Paris with 
his instruments and showed them the way. 
Hamilton was so well satisfied with the success of his new 
method in dealing with the problems presented by the propaga- 
tion of Light that full of enthusiasm he proceeded to apply a 
generalised form of the same method in the investigations of the 
motion of any material system, and a paper of his was read 
before the Royal Society in 1834 with the following title: ‘* On 
a general method in Dynamics by which the Study of the 
Motions of all free systems of attracting or repelling points is 
reduced to the Search and Differentiation of one Central 
Relation, or Characteristic Function.” 
, Fo show the importance attached by the most competent 
judges to Hamilton’s work in this field of Theoretical Dynamics, 
we cannot do better than quote the words of his great German 
contemporary Jacobi, who afterwards himself added to the new 
theory such valuable developments. 
Jacobi writes as follows :—‘‘ If a free system of material points 
is acted on by no other forces than such as arise from their 
mutual attraction or repulsion, the differential equations of their 
motion can be represented in a simple manner by means of the 
partial differential coefficients of a single function of the co- 
ordinates. Lagrange, who first made this important observa- 
tion, at the same time 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 integral equations of 
motion might be represented in just as simple a manner by means 
of the Partial Differential Coefficients 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 Section 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 which 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 Quaternion 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 Jeading 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 first set his own mind work- 
ing 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 systems 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 pro- 
posed for space of two dimensions the theory now known as that 
of the Complex Variable. For them «+ zy meant the vector to 
the point ay, and the product of two vectors meant a new vector 
of the same form, the only law required being that 7 operating 
upon 7 was always equivalent to —1. ; s 
Many attempts had been made to form on similar lines a Cal- 
culus 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 oné another leading to results hope- 
lessly involved. It was here that Hamilton’s wonderful faculty 
He proposed that 
