on the Principle of Least Pressure. 4 25 



maintained. It as these triangles approach evanescence they 

 continually approximate to a certain ratio or continually re- 

 tain the same ratio, and if this approximation or this per- 

 manence continue, however near we bring them to a state of 

 evanescence, it is a legitimate conclusion, that in their actual 

 state of evanescence they attain the ratio to which they have 

 continually approximated, or retain that which, up to the 

 point of evanescence, they have uniformly preserved. 



Now the forces have the ratio of the triangles in their finite 

 state, — this is a matter of demonstration, — and they retain this 

 ratio however nearly they are brought to the line AB ; when, 

 for instance, they are distant from that line by the smallest 

 conceivable quantity: it is therefore a legitimate conclusion 

 that, carrying on the approximation, their ultimate ratio is the 

 evanescent ratio of the triangles. 



My argument in respect to this matter of the three points 

 of support stands thus: I have shown (and Mr. Earnshaw 

 will not pretend to dispute my demonstration of this fact) that 

 when the points G and C are made to approach within the 

 smallest conceivable distance of the positions which I have ulti- 

 mately assigned them in the line AB, the pressures upon A,B 

 and C are each one third that upon C. 



If under these circumstances Mr. Earnshaw conceives that 

 these points G and C do not approach near enough to their 

 ultimate positions to be supposed to sustain pressures not very 

 different from those which they sustain when in those posi- 

 tions, then let the smallest conceivable spaces of which I have 

 spoken be halved, and let the points take up their positions 

 in the bisections of these spaces : the proposition will then be 

 true, and Mr. Earnshaw's objections will be limited within 

 the remaining half of the smallest space conceivable, where 

 I leave them. 



I now come to Mr. Earnshaw's attack upon my application 

 to his case of a formula derived from the principle of least 

 pressure, which formula, when thus applied, gives a result 

 identical with that just obtained by the method of evanescent 

 triangles, and thus furnishes a remarkable verification of the 

 principle of least pressure. I must confess that the objection 

 which Mr. Earnshaw has taken to my application of this for- 

 mula has perfectly astonished me. I have arrived at the equa- 

 tion 



x' (?) a — b — c) —2 x a (b \-ac — bc) = —abc, 



in which equation a l> and c are dependent upon the relative 

 positions of three p>inls of resistance situated anywhere in 

 Third Series. Vol. I. No. 24. June ls:5t. fl I 



