2 REPORT—1850. 
equations were obtained; and from these, X, Y, Z were determined. The result 
was very striking ; it was, that the probable error of the fixity of the instrument was 
ten times less than that of the stars, which, however, included the personal equation 
of the observer, errors of clock, transit telescope and others. The doubt, however, 
‘which assailed him was, whether he was justified in applying the doctrine of pro- 
babilities to obtain from those series the probable error of quantities which were not 
themselves mere results of actual observation, in which case there would be no doubt 
of its legitimate application, but where the quantities were, in each term of the series, 
partly the result of observation and partly deductions of formule. From this doubt 
he requested their assistance, either to extricate him or convince him of the incor- 
rectness of the method he used. Another question he begged to propose to the ma- 
thematicians for their assistance. It was well known that an equation of any order, 
containing only one unknown quantity, could be lowered one dimension, if one of its 
roots could be obtained numerically. Now,the question he wished to propose 
was, whether any mathematician knew any similar mode of lowering a system of 
several equations of high degree, which involved the same number of unknown but 
independent quantities ? 
On Polyzones inscribed on a Surface of the Second Order. 
By Sir W. R. Hamittron, MRA. 
On the Laws of the Elasticity of Solids. 
By W. J. Macquorn Rankine, CB, F.RASLE, FRSA. §e. 
This paper is intended to form the foundation of the theoretical part of a series of 
researches connected with the strength of materials. Its immediate object is to in- 
vestigate the relations which must exist between the elasticities of different kinds pos- 
sessed by a given substance, and between the different values of those elasticities in 
different directions. 
The different kinds of elasticity possessed by a solid substarice are distinguished 
into three :—First, longitudinal elasticity, representing the forces called into play in a 
given direction by condensation or dilatation of the particles of the body in the same 
direction. Secondly, lateral elasticity, representing those called into play in a given © 
direction by condensation or dilatation of the particles of the body in a direction at 
tight angles to that of the force; and thirdly, transverse elasticity, or rigidity, being 
the force by which solid substances resist distortion or change of figure, and the pro- 
perty which distinguishes solids from fiuids. The author’s researches refer chiefly 
to substances whose elasticity varies in different directions. His first endeavour is, 
to determine the laws of elasticity of such substances, so far as they are independent 
of hypotheses respecting the constitution of matter, a method which has not hitherto 
been followed. : ' 
The first theorem or law states the existence of three rectangular axes of elasticity 
in each substance possessing a certain degree of symmetry of molecular action. The 
elasticity of a body, as referred to tliese three axes; is expressed by twelve coefficients, 
three of longitudinal, and six of lateral elasticity, and three of rigidity, which are 
connected by the following laws :— 
Theorem Second.—The coefficient of rigidity is the same for all directions of dis- 
tortion in a given plane. ' 
Theorem Third.—In each of the coordinate planes of elasticity, the coefficient of 
rigidity is equal to one-fourth part of the sum of the two coefficients of longitudinal 
elasticity, diminished by one-fourth part of the sum of the two coefficients of lateral 
elasticity in the same plane. 
The investigation having now been carried as far as is practicable without the 
aid of hypotheses, the author determines, in the first place; the consequences of the 
supposition of Boscovich, that elasticity arises entirely from the mutual action οἵ. 
atomic centres of force. In the following theorems a perfect solid means a body so 
constituted. ? 
Theorem Fourth.—In each of the coordinate planes of elasticity of a perfect solid, 
the two coefticients of lateral elasticity and the coefficient of rigidity are all equal to” 
each other. 
