§ lO • OF ARTS AND SCIENCES. 399 



teristics which were anticipated by Professor Thomson; 

 and to prove by a second differentiation, as well as by ac- 

 tually plotting these curves, that the elasticity and its deriv- 

 ative do not more than twice pass through the value zero. 

 We have, therefore, every reason to expect that the true 

 form of the equation may have been found for the curves 

 whose characteristics were predicted thirteen years ago. 



If we suppose equations I. and II. to be simultaneously 

 equal to zero, we have, multiplying I. by 6, dividing 



P' 



both equations by — — , and transposing, 



6/0- (/q-O T^ (3/- 2/0 _ 36/0^ r^ , 5-1 



i^{i^i') 7; {i—V) /" L ^Po'J 



U (/q - o T (3/- 20- + w 36/0^ Ti -^ ^°1 

 P {I— I') 7; (/— O' ~ I' ^ ~^p:-^ 



whence 



P (/_/') 7; (/—/') i^{i—i')T, {i—iy 



Clearing effractions and common factors, 



6 (/- o (3/- 21') = (3/- 2/')^ + ir 

 18/^ _ 12/r — i8/r + 12/'^ = gp — 12/r + 4/'^ + ir 



9/^— 19/^ + 8/'^ = o 



72 19 zr I /I9V 7'2_ / 19 V 7'2 8 



9 



^ — 19 I I7Tq"V 8 



r 18 



± 



JBJ 



9 



Since / is essentially greater than /', only the positive 

 root is to be taken, and we have, finally, in the critical 

 state, the condition, 



Y— 1-53022 III. 



