of Edinburgh, Session 1873-74. 
335 
10. On the Stresses due to Compound Strains. By Prof. C. 
Niven. Communicated by Prof. Tait. 
{Abstract) 
In the treatment of questions which relate to the equilibrium 
vibrations of elastic solids, it is usual to suppose the substance to 
start from a state without strain, and in general to consider only 
the case of small distortions, for which squares and products of the 
space-variations of displacement may be neglected. The mathe- 
matical solution depends, in the first instance, on the expression 
of the work done in distorting an element. This, as was first done 
by Green, is expressed in terms of six functions of the distortions, 
termed strains. The part of the potential function which is of the 
second degree contains 21 coefficients, reducible to 18 by a proper 
choice of axes. But in the present state of our knowledge of the 
constitution of seolotropic substances, it is in general impossible to 
effect a further reduction, and it is probable that the function will 
have different forms according to the previous history of the 
substance. 
The case in which the eeolotropy has been produced by the action 
of considerable stress has formed the subject of investigations by 
Cauchy, St Yenant, and others. Cauchy’s results were based 
directly on the consideration of molecular attractions; and though 
the other authors have employed Green’s theory of the potential 
energy, they have still made use of molecules to find its form. 
In the present paper the author has sought to base the treatment 
of the subject on the law of superposition of one set of strains on 
another. If these states of strain be called respectively primary 
and secondary, it is shown that the total strains differ from the 
primary by linear functions of the secondary, and this whether the 
latter be small or large. The true form of the potential in terms 
of the secondary strains is thereafter readily found. It agrees so 
far with the result of M. Boussinesq, and furnishes expressions for 
the stresses agreeing to a certain extent with those originally given 
by Cauchy. 
There is one part of the potential energy due to the secondary 
strains which has not hitherto been discussed. It consists of terms 
