14 Mr. W. H. L. Russell on Linear Differential Equations. [Nov. 21, 



tion was not examined by means of the stethometer ; but from Mr. 

 Le Gros Clark's experiments it would appear that they have but little 

 influence in forced inspiration. 



With the intrinsic thoracic muscles the case is different. It is pro- 

 bable that, with the exception of the levatores costarum, each of these 

 muscles has some further power beyond that of simply raising or de- 

 pressing the ribs. 



The triangularis sterni is more especially a constrictor of the anterior 

 part of the thorax. 



The inter costals have very various actions assigned to them ; but when 

 all the evidence of Traube, Sibson, and others is considered, it seems 

 most probable that the six upper ribs are raised by the external inter- 

 costals, and depressed by the internal muscles ; but the ribs are not rigid 

 bars, and hence these muscles have a further action upon them. From 

 careful experiments upon a model with elastic vulcanite ribs, and 

 elastic bands stretched between them in the direction of (a) the external 

 intercostals, (b) the internal intercostals, and (c) with now the upper ribs 

 fixed, now the lower, it was concluded that whilst modifications were in- 

 troduced into their action by the last-named condition, yet that without 

 doubt the tendency of the external intercostals was, 1st, to draw the ribs 

 upwards ; 2nd, to separate their anterior ends ; 3rd, to straighten them. 

 On the other hand, the action of the internal intercostals was, 1st, to 

 draw the ribs downwards ; 2nd, to bring their anterior extremities nearer 

 together ; 3rd, to bend them inwards. These results were explained by 

 resolving the forces of these muscles in the directions (a) along the ribs, 

 (b) at right angles to them. 



The diaphragm may also be considered to have an important action in 

 bending the lower ribs. 



Evidence can be adduced showing the value of the stethometer used in 

 these inquiries, both in physiological studies and in medical practice. 



III. " On Linear Differential Equations." — No. VI. 

 By W. H. L. Russell, E.R.S. Received July 30, 1872. 



We now consider linear differential equations which are satisfied by the 

 roots of an algebraical equation admitting of explicit solution. To de- 

 termine in what cases the linear differential equation 



is satisfied by assuming 



when X, Y, Z are rational functions of (pc). We shall commence by 



