540 Viswjo*^! \i's.i^^s\ Royal Society^i*r^K vAUlL ,to'j'! 



types of refepenee aite respectively (/, in, », Mifh^ Im^nip, •*'» ^^'^ 



X', /i', v'), we have li .rTnioi^fiimuuV) ')i{'V 



')iii 1 I » cos0=//' + »im'4-w«' + XX'-f /ifi' + i'v'. 

 "^ tli^'tifeatment of the subject iu the text of the paper is quite 

 abstract, but along with it a series of examples are given, illustrating 

 the statements by applications to familiar types of stresses and 

 'Strains. 



^^' Part II. commences with an interpretation of the Differential 



^Equation of the potential energy of Elasticity of a solid, in terms of the 



«ihode of specification of stresses and strains laid down in Part I. 



The Quadratic Function expressing the potential energy of an elastic 



solid when strained to an infinitely small amount, is next considered ; 



and its simplest possible form, that of six squares with coefficients, is 



•interpreted. Hence it is proved that an infinite number of systems 



^k^ six types of strains or sti'esses exist in any given elastic solid^ 



such that if a strain of any one of those types be impressed on the 



hodyy the elastic reaction is balanced by a stress orthogonal to the 



five others of the same system. 



It is next shown that there is necessarily one, and in general only 

 one, such system of six types of strain for an elastic solid which are 

 all mutually orthogonal ; and the types belonging to this system are 

 called the Six Principal Strain Types of the body. 

 ^ ' The characteristic of a Principal Strain Type is, that the stress re- 

 quired to keep a body in a state of strain of such a type, is of the same 

 type as the strain. The six Principal Elasticities of a body are the 

 six coefficients by which strains of the six Principal Types must be 

 multiplied to find the stress required to maintain them. 

 xfoi.In conclusion, reasons are given for believing that natural crystals 

 may exist for which there are six unequal Principal Elasticities, and 

 consequently six different, and only six different, Principal Strain- 

 types. 



A corollary regarding the property which certain liquids and cry- 

 stals possess of causing a rotation in the plane of polarization of light 

 passing through them, and Paraday's optical property of transparent 

 y]l)odies under magnetic force, is inferred, and is more fully considery^d 

 \jPi a subsequent communication to the Royal Society. ,',^ 



9fi ' ** On the Construction of the Imperial Standard Pound, and its 

 hCJopies of Platinum ; and on the comparison of the Imperial Standard 

 liPound with the Kilogramme des Archives." By W. H. Miller, M.A., 

 of.R.S., Professor of Mineralogy jn the University of Cambridge. — 



i6) ^^t'Qtkrttimsioncrs appointed in i 1838 ftdiconfeidwf' the s*epfl to be 



-'♦aken for the restoration of the standards of weight and measure, to 



»lr*plaee those which were destroyed by the burning of the Houses of 



-OJ^iriifiinldBt, found provisions for the restoration of the lost standards 



prescribed to them by Sections 3 and 5 of the Act 5th George IV., 



whereby it is dire6ted that, in case of the loss of the standards, the 



yard shall be restored by takina tlic length which shall bear a certain 



relation to the length of the ])( ndnliun, vibrating seconds of mean 



time, in « vacuum; at the level of the sea ; and that the pound shall 



be restored by taking the weight which bears a certain proportion to 



