4 REPORT—1843., 
bility being therefore in this case only 6~®. The advantage of the method is that 
what has here been called the argument of probability depends, in general, more simply 
than the probability itself on the conditions of a question; while the introduction of 
this new conception and nomenclature allows some of the most important known re- 
sults respecting the mean results of many observations to be enunciated in a simple 
and elegant manner. 

A paper was read on some Investigations connected with Equations of the Fifth 
Degree, by Sir W. Hamilton. 

On the Method of Graphical Representation, as applied to Physical Results. 
By Professor Lioyn. 
It is well known that if a series of ordinates be taken to denote the observed values 
of any physical quantities, the corresponding abscissz denoting the respective values 
of the variable upon which it depends, the course of the first variable, at intermediate 
points, may be represented by drawing a curve through the extremities of the ordinates 
of observation, the exactness of the representation depending on the shortness of the 
intervals. The observed values of the ordinates, however, being subject to the errors 
of observation, it is manifest that their extremities are not necessarily points of the 
representative curve; and the object of the author was to inquire whether, and under 
what circumstances, other points could be substituted for those of immediate observa- 
tion, the former being connected with the latter by known relations. Such a course 
is usually resorted to, when the intervals of observation are small, and the errors con- 
siderable, the curve being in such cases drawn (not through, but) among the points 
furnished by observation, allowing a weight in proportion to their number. 
The validity of this process appears to depend on two principles, viz. first, that the 
positive and negative errors are equally probable; and second, the assumption that the 
function represented is not subject to abrupt changes. It is obvious, however, that its 
applicability in any particular case will depend upon the relation which subsists be- 
tween the intervals of the successive ordinates and their probable errors; and it is 
important to know what that condition is. The points connected with those of observa- 
tion by the simplest relations, are those obtained by bisecting the interval between 
each successive pair, or taking the arithmetical mean both of the ordinates and ab- 
scisse. It is very easy to express in this case the relation sought. If f(x) denote the 
value of the function represented, corresponding to the abscissa a, and if x — h denote 
the preceding value of the latter, then it is obvious that the error committed by taking 
the arithmetical mean of f(x) and f(a — h), for the ordinate corresponding to the ab- 
scissa « — th, will be represented by the series fs 
h? 
P@) re =I 1.2.3 5 Bias 
in which, in general, the first term may be made to surpass the sum of all the rest, by 
taking h sufficiently small. On the other hand, the error saved by the substitution 
of the arithmetical mean of the observed ordinates is 
‘(1-Js), 
e denoting the probable error of a single observation; and the process in question will 
be advantageous when the former of these quantities is /ess than the latter. It thus 
appears that the condition of its applicability may be expressed by the following rela- 
tion between / and e: V2(VI1)e 
P< PPA gg OT 

On the Theory of Total Reflexion, and of the Insensible Refraction which 
accompanies it. By Professor MAcCuLLaGu. 
The phznomenon of total reflexion has for a long period excited the attention of ma- 

