on the Principle of Least Pressure. 421 



out of the question in taking their sum. On what, again, is 

 this conclusion grounded ? It is true that I have taken the 

 sum of the resistances each with a positive sign, and asserted 

 it to be a minimum ; but I have also asserted each to be an 

 independent function of the coordinates of its point of appli- 

 cation, so that a minimum value of the sum of the functions 

 supposes a minimum value of each function in particular; and 

 conversely. 



In fact, the conclusion that the sum of the resistances is a 

 minimum is consequent upon the assumed minimum value of 

 each resistance in particular, (as is plain enough from the 

 order of my demonstration,) and was merely used by me as 

 affording a more direct and comprehensive method of ex- 

 pressing the conditions of the question in the language of 

 analysis*. 



Mr. Earnshaw's first and great argument, that on which he 

 states himself to place his principal reliance, is this: " It ap- 

 pears to me," says he, " that, speaking generally, it is impos- 

 sible that each force of the system C should sustain only such 

 pressures as are propagated to it from the system A ; unless 

 either that the system C consists of but one force, or that all 

 the forces of which it consists are parallel. For if there be 

 more forces than one in the system C, and if they are not 

 parallel, their actions must of necessity mutually propagate 

 pressures." 



Now here I at once join issue with Mr. Earnshaw, and 

 assert, that it is possible, &c. &c. ; and I give him this case in 

 proof of the possibility, which, be it observed, is the point in 

 question. " Suppose that to the different points of applica- 

 tion of the system A (having first removed that system) we 

 apply forces having for their resultant one of the forces C; 

 and then to the same points forces having for their resultant 

 another of the forces of the system C, and so on, until we have 

 obtained all the forces of the system C. Now, under these 



* There is one case in which this assumption of a minimum value of the 

 sum of the resistances consequent upon the assumed minimum values of 

 the several component resistances fails : this assumption, as stated above 

 I have made, however, only as a convenient enunciation of my fundamental 

 proposition. The case to which I have alluded is that of a system of pa- 

 rallel resistances. The sum of these resistances is not a minimum subject to 

 the conditions. But still the original proposition holds, each force in parti- 

 cular is a minimum; subject (of course) to the condition that every other 

 is a minimum, and their sum a constant. Under these circumstances the 

 analytical investigation of this particular case is included in that which I 

 have given of the general proposition. This is precisely the case which 

 Mr. Earnshaw has brought forward in the shape of a new objection to my 

 theory. I think it better to confine myself to his three original objections, 

 until they are completely disposed of; I shall then be happy to meet him 

 on this. 



