164 MEMOIR OF THE LATE PROF. E. HODGKINSON^ F.R.S. 



rods are taken into account. The introduction of these 

 complex, though necessary, elements into the question, 

 led to the formation of the following difficult and compre- 

 hensive differential equation : — 



ctdor I 



--^T:zhz-\rcy-\-e\xdy .... (A), 



where x^ y, are the current coordinates of a point in the 

 curve, and z the length of the curve from this point to 

 the lowest point. The explanation of the constants «, h, c, e 

 is as follows : — 



fi! = the tension of the curve at its lowest point. 



^=the weight of a unit of length of the curve. 



c=the weight of a unit of length in the roadway, which is 

 supposed to be divided transversely into separate parts, 

 and may include any weight uniformly distributed 

 over it, with that of the suspension-rods below the 

 horizontal line. 



e=the weight of a unit of vertical surface in the suspend- 

 ing-rods, the rods being here supposed to be uniformly 

 distributed, and indefinitely near to one another, and 

 therefore reckoned as a uniform surface. 



This diflPerential equation, under given conditions of the 

 constants, is treated in this paper in a very able manner, 

 showing great command over the resources of modern 

 analysis, and facility in the use of the varied artifices em- 

 ployed in the integration of difierential equations. The 

 results arrived at have been referred to by the ablest writers 

 of the age. Dr. Whewell and the Eev. Canon Moseley, — 

 by the former in his ' Analytical Statics,^ where the solu- 

 tion of equation (A), as given by Mr. Hodgkinson, occupies 

 a distinguished place ; by the latter in his ^ Engineering 

 and Architecture,^ in which the labours of our late friend 

 are honourably mentioned : — 



" This problem appears first to have been investigated 



